Original: We know that the Ravens had a touchdown
Revised: We know that the Ravens had a touchdown

Original: and a ﬁeld goal at
Revised: and a ﬁeld goal at

Original: halftime.
Revised: halftime.

Original: We
Revised: We

Original: also know
Revised: also know

Original: that a touchdown is worth 7 points and a ﬁeld goal is worth 3 points.
Revised: that a touchdown is worth 7 points and a ﬁeld goal is worth 3 points.

Original: 7 + 3 = 10.
Revised: 7 + 3 = 10.

Original: The answer is
Revised: The answer is

Original: The Vikings ﬂew to Bank of America Stadium to face the Carolina Panthers.
Revised: The Vikings ﬂew to Bank of America Stadium to face the Carolina Panthers.

Original: After a scoreless ﬁrst quarter, Carolina got on the board with quarterback Matt Moore ﬁnding fullback Brad Hoover on a 1-yard TD pass.
Revised: After a scoreless ﬁrst quarter, Carolina got on the board with quarterback Matt Moore ﬁnding fullback Brad Hoover on a 1-yard TD pass.

Original: After yet another scoreless quarter, Carolina sealed the game as Matt Moore completed a 42-yard touchdown pass to wide receiver Steve Smith.
Revised: After yet another scoreless quarter, Carolina sealed the game as Matt Moore completed a 42-yard touchdown pass to wide receiver Steve Smith.

Original: Q: What
Revised: Q: What

Original: were all the
Revised: were all the

Original: scoreless quarters?
Revised: scoreless quarters?

Original: A:
Revised: A:

Original: The answer is: the ﬁrst quarter and the third quarter.
Revised: The answer is: the ﬁrst quarter and the third quarter.

Original: Q: How many scoreless quarters were there?
Revised: Q: How many scoreless quarters were there?

Original: A:
Revised: A:

Original: We know that the ﬁrst and third quarters were all the scoreless quarters.
Revised: We know that the ﬁrst and third quarters were all the scoreless quarters.

Original: The answer is 2.
Revised: The answer is 2.

Original: 8. E XAMPLES WHERE LEAST - TO - MOST SUCCEEDED BUT CHAIN - OF - THOUGHT FAILED 8.4. C ASE
Revised: 8. E XAMPLES WHERE LEAST - TO - MOST SUCCEEDED BUT CHAIN - OF - THOUGHT FAILED 8.4. C ASE

Original: Q: The ﬁrst two Buick automobiles were made in 1899 and 1900 at the “Buick Auto-Vim and Power
Revised: Q: The ﬁrst two Buick automobiles were made in 1899 and 1900 at the “Buick Auto-Vim and Power

Original: Company” by chief-engineer Walter Lorenzo Marr, but company owner David Dunbar Buick was reluctant to begin making automobiles, being satisﬁed with stationary and marine engine production, so Marr left Buick in 1901 to found Marr (automobile) under his own name.
Revised: Company” by chief-engineer Walter Lorenzo Marr, but company owner David Dunbar Buick was reluctant to begin making automobiles, being satisﬁed with stationary and marine engine production, so Marr left Buick in 1901 to found Marr (automobile) under his own name.

Original: His replacement was
Revised: His replacement was

Original: Eugene Richard, who applied for a patent in 1902 for Marrs valve-in-head (Overhead valve engine) engine, which patent, number 771,095, was awarded to Richard in the name of Buick in 1904.
Revised: Eugene Richard, who applied for a patent in 1902 for Marrs valve-in-head (Overhead valve engine) engine, which patent, number 771,095, was awarded to Richard in the name of Buick in 1904.

Original: In 1903, the third Buick automobile was made, this time by Richard, but in 1904 Buick, whose company was now called “Buick Motor Company,” moved to Flint, Michigan, and Richard stayed behind.
Revised: In 1903, the third Buick automobile was made, this time by Richard, but in 1904 Buick, whose company was now called “Buick Motor Company,” moved to Flint, Michigan, and Richard stayed behind.

Original: Marr was rehired in Flint as chief engineer, to begin making automobiles in production.
Revised: Marr was rehired in Flint as chief engineer, to begin making automobiles in production.

Original: That year, 37 Buick automobiles were made, production increasing to 750 in 1905, 1,400 in 1906, 4, in 1907, and 8,800 in 1908, taking the number one spot away from close competitors Oldsmobile,
Revised: That year, 37 Buick automobiles were made, production increasing to 750 in 1905, 1,400 in 1906, 4, in 1907, and 8,800 in 1908, taking the number one spot away from close competitors Oldsmobile,

Original: Ford Motor Company, and Maxwell automobile.
Revised: Ford Motor Company, and Maxwell automobile.

Original: How many Buick automobiles were built between 1904 and 1905?
Revised: How many Buick automobiles were built between 1904 and 1905?

Original: Chain-of-thought by code-davinci-002 A:
Revised: Chain-of-thought by code-davinci-002 A:

Original: According to the passage, 37 Buick automobiles were made in 1904, and 750 were made in 1905.
Revised: According to the passage, 37 Buick automobiles were made in 1904, and 750 were made in 1905.

Original: 750 - 37 = 713.
Revised: 750 - 37 = 713.

Original: So the answer is 713.
Revised: So the answer is 713.

Original: Least-to-most by code-davinci-002 Q: How many Buick automobiles were built in 1904?
Revised: Least-to-most by code-davinci-002 Q: How many Buick automobiles were built in 1904?

Original: A: “In 1903, the third Buick automobile was made, this time by Richard, but in 1904 Buick, whose company was now called “Buick Motor Company,” moved to Flint, Michigan, and Richard stayed behind.
Revised: A: “In 1903, the third Buick automobile was made, this time by Richard, but in 1904 Buick, whose company was now called “Buick Motor Company,” moved to Flint, Michigan, and Richard stayed behind.

Original: Marr was rehired in Flint as chief engineer, to begin making automobiles in production.
Revised: Marr was rehired in Flint as chief engineer, to begin making automobiles in production.

Original: That year, 37 Buick automobiles were made”.
Revised: That year, 37 Buick automobiles were made”.

Original: So the answer is 37.
Revised: So the answer is 37.

Original: Q: How many Buick automobiles were built in 1905?
Revised: Q: How many Buick automobiles were built in 1905?

Original: A: “production increasing to 750 in 1905”.
Revised: A: “production increasing to 750 in 1905”.

Original: So the
Revised: So the

Original: answer is 750.
Revised: answer is 750.

Original: Q
Revised: Q

Original: : How
Revised: : How

Original: many Buick automobiles were built between 1904 and 1905?
Revised: many Buick automobiles were built between 1904 and 1905?

Original: A: We know that 37 Buick automobiles were built in 1904.
Revised: A: We know that 37 Buick automobiles were built in 1904.

Original: We also know that 750 Buick automobiles were built in 1905.
Revised: We also know that 750 Buick automobiles were built in 1905.

Original: So 37 + 750 = 

Revised: So 37 + 750 =

Original: Buick automobiles were built between 1904 and 1905.
Revised: Buick automobiles were built between 1904 and 1905.

Original: So the answer is 787.
Revised: So the answer is 787.

Original: Analysis Chain-of-thought failed because it computed “750 - 37 = 713”, which is wrong.
Revised: Analysis Chain-of-thought failed because it computed “750 - 37 = 713”, which is wrong.

Original: The correct answer is “37 + 750 = 787”.
Revised: The correct answer is “37 + 750 = 787”.

Original: 8.4. C ASE
Revised: 8.4. C ASE

Original: Q: The Guatemalan authorities were unaware of ongoing contacts between the colonial authorities in
Revised: Q: The Guatemalan authorities were unaware of ongoing contacts between the colonial authorities in

Original: Yucat´an and the Itza.
Revised: Yucat´an and the Itza.

Original: In the second half of 1695 President Barrios began planning another expedition against the Itza from Verapaz, following harsh criticism of Juan D´ıaz de Velasco’s retreat from near
Revised: In the second half of 1695 President Barrios began planning another expedition against the Itza from Verapaz, following harsh criticism of Juan D´ıaz de Velasco’s retreat from near

Original: Lake Pet´en Itz´a in April of that year.
Revised: Lake Pet´en Itz´a in April of that year.

Original: Barrios died in November, and oidor Jos´e de Escals became acting head of the Guatemalan colonial administration.
Revised: Barrios died in November, and oidor Jos´e de Escals became acting head of the Guatemalan colonial administration.

Original: He appointed oidor Bartolom´e de Am´esqueta to lead the next expedition against the Itza.
Revised: He appointed oidor Bartolom´e de Am´esqueta to lead the next expedition against the Itza.

Original: Am´esqueta marched his men through torrential rain from Cahab´on to Mop´an, arriving on 25 February 1696; the Guatemalans were still unaware of the clashes between Garc´ıa’s forces and the Itza.
Revised: Am´esqueta marched his men through torrential rain from Cahab´on to Mop´an, arriving on 25 February 1696; the Guatemalans were still unaware of the clashes between Garc´ıa’s forces and the Itza.

Original: Am´esqueta’s expedition was suffering from shortages of native labour and supplies.
Revised: Am´esqueta’s expedition was suffering from shortages of native labour and supplies.

Original: Twenty-ﬁve of his 150 men were sick, and his advance stalled as the force rested up at Mop´an.
Revised: Twenty-ﬁve of his 150 men were sick, and his advance stalled as the force rested up at Mop´an.

Original: Captain D´ıaz de Velasco volunteered to lead a party of 25 men ahead to the lake; he was accompanied by Dominican friars Crist´obal de Prada and Jacinto de Vargas.
Revised: Captain D´ıaz de Velasco volunteered to lead a party of 25 men ahead to the lake; he was accompanied by Dominican friars Crist´obal de Prada and Jacinto de Vargas.

Original: He was also accompanied by AjK’ixaw, the Itza nobleman who had been taken prisoner on D ´
Revised: He was also accompanied by AjK’ixaw, the Itza nobleman who had been taken prisoner on D ´

Original: ıaz’s previous expedition, who served as a trusted guide, scout and translator.
Revised: ıaz’s previous expedition, who served as a trusted guide, scout and translator.

Original: They set out from Mop´an on 7 March and left some sick soldiers with some supplies after ﬁve days, they then caught up with an advance party of roadbuilders and their military escort at R´ıo Chakal.
Revised: They set out from Mop´an on 7 March and left some sick soldiers with some supplies after ﬁve days, they then caught up with an advance party of roadbuilders and their military escort at R´ıo Chakal.

Original: The two forces combined, giving D´ıaz a force of 49 soldiers and 34 porters and archers from Verapaz.
Revised: The two forces combined, giving D´ıaz a force of 49 soldiers and 34 porters and archers from Verapaz.

Original: Once they arrived at IxB’ol, near the shore of Lake Pet´en Itz´a, AjK’ixaw was sent ahead as an emissary to Nojpet´en.
Revised: Once they arrived at IxB’ol, near the shore of Lake Pet´en Itz´a, AjK’ixaw was sent ahead as an emissary to Nojpet´en.

Original: How many months after Juan D´ıaz de Velasco’s retreat did President Barrios die?
Revised: How many months after Juan D´ıaz de Velasco’s retreat did President Barrios die?

Original: Chain-of-thought by code-davinci-002 A:
Revised: Chain-of-thought by code-davinci-002 A:

Original: According to the passage, President Barrios died in November, and the expedition against the Itza was led by Bartolom´e de Am´esqueta in February.
Revised: According to the passage, President Barrios died in November, and the expedition against the Itza was led by Bartolom´e de Am´esqueta in February.

Original: So the answer is 3.
Revised: So the answer is 3.

Original: Least-to-most
Revised: Least-to-most

Original: by code-
Revised: by code-

Original: davinci
Revised: davinci

Original: -002 Q: When did
Revised: -002 Q: When did

Original: Juan D´ıaz de Velasco’s
Revised: Juan D´ıaz de Velasco’s

Original: retreat?
Revised: retreat?

Original: A: “April of that year”.
Revised: A: “April of that year”.

Original: So the answer is April.
Revised: So the answer is April.

Original: Q: When did President Barrios die?
Revised: Q: When did President Barrios die?

Original: A: “November”.
Revised: A: “November”.

Original: So the answer is November.
Revised: So the answer is November.

Original: Q: How many months after Juan D´ıaz de Velasco’s retreat did President Barrios die?
Revised: Q: How many months after Juan D´ıaz de Velasco’s retreat did President Barrios die?

Original: A:
Revised: A:

Original: We
Revised: We

Original: know that
Revised: know that

Original: Juan D´ıaz de Velasco’s
Revised: Juan D´ıaz de Velasco’s

Original: retreat
Revised: retreat

Original: was
Revised: was

Original: in April
Revised: in April

Original: We also know that President Barrios
Revised: We also know that President Barrios

Original: died
Revised: died

Original: in November.
Revised: in November.

Original: So November - April = 7 months after Juan D´ıaz de Velasco’s retreat
Revised: So November - April = 7 months after Juan D´ıaz de Velasco’s retreat

Original: did
Revised: did

Original: President
Revised: President

Original: Barrios die.
Revised: Barrios die.

Original: So
Revised: So

Original: the
Revised: the

Original: answer
Revised: answer

Original: is 7.
Revised: is 7.

Original: Analysis Chain-of-thought generated the wrong ﬁnal answer because one fact it used, “the expedition against the Itza was led by Bartolom´e de Am´esqueta in February”, is unrelated.
Revised: Analysis Chain-of-thought generated the wrong ﬁnal answer because one fact it used, “the expedition against the Itza was led by Bartolom´e de Am´esqueta in February”, is unrelated.

Original: 8.4. C ASE
Revised: 8.4. C ASE

Original: Q: Peele was christened on 25 July 1556 at St James Garlickhythe in the City of London.
Revised: Q: Peele was christened on 25 July 1556 at St James Garlickhythe in the City of London.

Original: His father,
Revised: His father,

Original: James Peele (d. Dec 30, 1585), who appears to have belonged to a Devonshire family, was clerk of Christs Hospital, a school which was then situated in central London, and wrote two treatises on bookkeeping, The Maner and Fourme
Revised: James Peele (d. Dec 30, 1585), who appears to have belonged to a Devonshire family, was clerk of Christs Hospital, a school which was then situated in central London, and wrote two treatises on bookkeeping, The Maner and Fourme

Original: How to Kepe a Perfecte Reconyng (1553)
Revised: How to Kepe a Perfecte Reconyng (1553)

Original: and The Pathe Waye to Perfectnes (1569).
Revised: and The Pathe Waye to Perfectnes (1569).

Original: The latter depicts James Peele in a woodcut on the title page.
Revised: The latter depicts James Peele in a woodcut on the title page.

Original: No contemporary likeness of George is known, although he was said to be short-legged, dark complected, red haired, and squinting, although not necessarily from reliable sources.
Revised: No contemporary likeness of George is known, although he was said to be short-legged, dark complected, red haired, and squinting, although not necessarily from reliable sources.

Original: Georges mother, Anne, died July 1, 1580, and his father married Christian Widers (d. 1597 in St. Christopher le Stocks) on November 3, 1580.
Revised: Georges mother, Anne, died July 1, 1580, and his father married Christian Widers (d. 1597 in St. Christopher le Stocks) on November 3, 1580.

Original: She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell.
Revised: She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell.

Original: His siblings included Anne (d. Jan 10, 1568/9), Isabel,
Revised: His siblings included Anne (d. Jan 10, 1568/9), Isabel,

Original: Judith (d. Apr. 16, 1582), and James (b. Jan 3, 1563/4).
Revised: Judith (d. Apr. 16, 1582), and James (b. Jan 3, 1563/4).

Original: Anne married John Alford on May 14, 1565, and had one son, Robert (October 9, 1567- c. March 12, 1654/5).
Revised: Anne married John Alford on May 14, 1565, and had one son, Robert (October 9, 1567- c. March 12, 1654/5).

Original: Judith married John Jackman on
Revised: Judith married John Jackman on

Original: June 19, 1575, and had three children, Susan (b. June 3, 1576), William (April 30, 1577-July 1, 1577), and Sarah (d. May 24, 1578).
Revised: June 19, 1575, and had three children, Susan (b. June 3, 1576), William (April 30, 1577-July 1, 1577), and Sarah (d. May 24, 1578).

Original: On Feb. 5, 1568/9, Isabel married Mathew Shakespeare, with whom she had eight children.
Revised: On Feb. 5, 1568/9, Isabel married Mathew Shakespeare, with whom she had eight children.

Original: Many scholars believe that this was a cousin of William Shakespeare, but this has not been veriﬁed.
Revised: Many scholars believe that this was a cousin of William Shakespeare, but this has not been veriﬁed.

Original: James Peele also wrote the Ironmongers Pageants of 1566 and 1569, which may have led to Georges writing of two Lord Mayors pageants.
Revised: James Peele also wrote the Ironmongers Pageants of 1566 and 1569, which may have led to Georges writing of two Lord Mayors pageants.

Original: How many years after her marriage to James Peele did Christian Widers die?
Revised: How many years after her marriage to James Peele did Christian Widers die?

Original: Chain-of-thought by code-davinci-002 A:
Revised: Chain-of-thought by code-davinci-002 A:

Original: According to the passage, Christian Widers died in 1597.
Revised: According to the passage, Christian Widers died in 1597.

Original: James Peele died in 1585.
Revised: James Peele died in 1585.

Original: 1597 - 1585 = 12.
Revised: 1597 - 1585 = 12.

Original: So the answer is 12.
Revised: So the answer is 12.

Original: Least-to-most by code-davinci-002 Q: When did Christian Widers marry James Peele?
Revised: Least-to-most by code-davinci-002 Q: When did Christian Widers marry James Peele?

Original: A: “She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell”.
Revised: A: “She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell”.

Original: So the answer is 1580.
Revised: So the answer is 1580.

Original: Q: When did Christian Widers die?
Revised: Q: When did Christian Widers die?

Original: A: “She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell”.
Revised: A: “She became a nurse on the Hospital payroll, where she remained ﬁve years after James Peeles death, when she married Ralph Boswell”.

Original: So the answer is 1597.
Revised: So the answer is 1597.

Original: Q: How many years after her marriage to James Peele did Christian Widers die?
Revised: Q: How many years after her marriage to James Peele did Christian Widers die?

Original: A: We know that Christian Widers married James Peele in 1580.
Revised: A: We know that Christian Widers married James Peele in 1580.

Original: We also know that Christian Widers died in 1597.
Revised: We also know that Christian Widers died in 1597.

Original: So 1597 - 1580 = 17 years after her marriage to James Peele did Christian Widers die.
Revised: So 1597 - 1580 = 17 years after her marriage to James Peele did Christian Widers die.

Original: So the answer is 17.
Revised: So the answer is 17.

Original: Analysis
Revised: Analysis

Original: 8.4. C ASE
Revised: 8.4. C ASE

Original: Q: Among the other denominations there are approximately 93,000 Mormons in 253 congregations, 25,000 Jewish adherents in 21 synagogues, 12,000 Muslims in 39 masjids, 7,000 Buddhists intemples, 7,000 Hindus in 17 temples, 2,500 Unitarian Universalism in 9 congregations, 2,000 Bahai in 17 temples, 5 Sikh temples, a Zoroastrian temple, a Jain temple and an uncounted number of neopagans.
Revised: Q: Among the other denominations there are approximately 93,000 Mormons in 253 congregations, 25,000 Jewish adherents in 21 synagogues, 12,000 Muslims in 39 masjids, 7,000 Buddhists intemples, 7,000 Hindus in 17 temples, 2,500 Unitarian Universalism in 9 congregations, 2,000 Bahai in 17 temples, 5 Sikh temples, a Zoroastrian temple, a Jain temple and an uncounted number of neopagans.

Original: How many more Mormons were there compared to Muslims?
Revised: How many more Mormons were there compared to Muslims?

Original: Chain-of-thought by code-davinci-002 A:
Revised: Chain-of-thought by code-davinci-002 A:

Original: According to the passage, there are 93,000 Mormons and 25,000 Muslims.
Revised: According to the passage, there are 93,000 Mormons and 25,000 Muslims.

Original: 93,000 - 25,000 = 68,000.
Revised: 93,000 - 25,000 = 68,000.

Original: So the answer is 68,000.
Revised: So the answer is 68,000.

Original: Least-to-most by code-davinci-002 Q: How many Mormons were there?
Revised: Least-to-most by code-davinci-002 Q: How many Mormons were there?

Original: A: “93,000 Mormons”.
Revised: A: “93,000 Mormons”.

Original: So the answer is 93,000.
Revised: So the answer is 93,000.

Original: Q: How many Muslims were there?
Revised: Q: How many Muslims were there?

Original: A: “12,000 Muslims”
Revised: A: “12,000 Muslims”

Original: So the
Revised: So the

Original: answer is
Revised: answer is

Original: Q: How many more Mormons were there compared to Muslims?
Revised: Q: How many more Mormons were there compared to Muslims?

Original: A: We know that there were 93,000 Mormons.
Revised: A: We know that there were 93,000 Mormons.

Original: We also know that there were 12,000 Muslims.
Revised: We also know that there were 12,000 Muslims.

Original: So there were 93,000 - 12,000 = 81,000 more Mormons than Muslims.
Revised: So there were 93,000 - 12,000 = 81,000 more Mormons than Muslims.

Original: So the answer is 81,000.
Revised: So the answer is 81,000.

Original: Analysis Chain-of-thought failed because it got one intermediate fact (the number of Muslims) wrong.
Revised: Analysis Chain-of-thought failed because it got one intermediate fact (the number of Muslims) wrong.

Original: 8.4. C ASE
Revised: 8.4. C ASE

Original: Q: Killigrew lived in Hanworth in Middlesex and Falmouth in Cornwall.
Revised: Q: Killigrew lived in Hanworth in Middlesex and Falmouth in Cornwall.

Original: On 4 November
Revised: On 4 November

Original: Killigrew married in the church of St Peter Le Poer, London, Catherine, fourth daughter of Sir
Revised: Killigrew married in the church of St Peter Le Poer, London, Catherine, fourth daughter of Sir

Original: Anthony Cooke.
Revised: Anthony Cooke.

Original: He thus became Cecil’s brother-in-law.
Revised: He thus became Cecil’s brother-in-law.

Original: His wife died in 1583, and on 7 November 1590 he was married in the same church to Ja ´ el de Peigne, a French Hugenot.
Revised: His wife died in 1583, and on 7 November 1590 he was married in the same church to Ja ´ el de Peigne, a French Hugenot.

Original: She was naturalised in June 1601.
Revised: She was naturalised in June 1601.

Original: After Henry’s death she remarried on 19 April 1617 George Downham, Bishop of
Revised: After Henry’s death she remarried on 19 April 1617 George Downham, Bishop of

Original: Derry, and died c.1632.
Revised: Derry, and died c.1632.

Original: By his ﬁrst wife, Killigrew had
Revised: By his ﬁrst wife, Killigrew had

Original: four
Revised: four

Original: daughters: How many years after here marriage
Revised: daughters: How many years after here marriage

Original: to Killigrew
Revised: to Killigrew

Original: did
Revised: did

Original: Catherine
Revised: Catherine

Original: die?
Revised: die?

Original: Chain-of-thought by code-davinci-002 A:
Revised: Chain-of-thought by code-davinci-002 A:

Original: According to the passage, Catherine died in 1583, and Killigrew married Ja ´ el de Peigne in 1590.
Revised: According to the passage, Catherine died in 1583, and Killigrew married Ja ´ el de Peigne in 1590.

Original: 1590 - 1583 = 7.
Revised: 1590 - 1583 = 7.

Original: So the answer is 7.
Revised: So the answer is 7.

Original: Least-to-most by code-davinci-002 Q: When did Catherine marry Killigrew?
Revised: Least-to-most by code-davinci-002 Q: When did Catherine marry Killigrew?

Original: A: “On 4 November 1566 Killigrew married in the church of St Peter Le Poer, London, Catherine”.
Revised: A: “On 4 November 1566 Killigrew married in the church of St Peter Le Poer, London, Catherine”.

Original: So the answer is 4 November 1566.
Revised: So the answer is 4 November 1566.

Original: Q
Revised: Q

Original: When did Catherine die?
Revised: When did Catherine die?

Original: A
Revised: A

Original: : “His wife died in 1583”.
Revised: : “His wife died in 1583”.

Original: So the answer is
Revised: So the answer is

Original: Q: By his ﬁrst wife, Killigrew had four daughters: How many years after here marriage to Killigrew did Catherine die?
Revised: Q: By his ﬁrst wife, Killigrew had four daughters: How many years after here marriage to Killigrew did Catherine die?

Original: A:
Revised: A:

Original: We know that Catherine married Killigrew in
Revised: We know that Catherine married Killigrew in

Original: We also know that Catherine died in
Revised: We also know that Catherine died in

Original: So 1583 - 1566 = 17 years after here marriage to Killigrew did Catherine die.
Revised: So 1583 - 1566 = 17 years after here marriage to Killigrew did Catherine die.

Original: So the answer is 17.
Revised: So the answer is 17.

Original: Analysis Chain-of-thought generated the wrong ﬁnal answer because one fact it used, “Killigrew married Ja´el de Peigne in 1590”, is unrelated.
Revised: Analysis Chain-of-thought generated the wrong ﬁnal answer because one fact it used, “Killigrew married Ja´el de Peigne in 1590”, is unrelated.

Original: 8. E RROR ANALYSIS : L EAST - TO - MOST PROMPTING
Revised: 8. E RROR ANALYSIS : L EAST - TO - MOST PROMPTING

Original: We randomly picked 20 failure cases, and found that out of those 20 failure cases:
Revised: We randomly picked 20 failure cases, and found that out of those 20 failure cases:

Original: • 4 are due to wrong problem decomposition – the decomposed problems do not make sense, or there is no decomposition at all.
Revised: • 4 are due to wrong problem decomposition – the decomposed problems do not make sense, or there is no decomposition at all.

Original: • 13 are due to wrong problem solving – it gave the wrong answer for a decomposed problem.
Revised: • 13 are due to wrong problem solving – it gave the wrong answer for a decomposed problem.

Original: • 3 are due to wrong “ground truth” – the given label is wrong regardless whether the prediction is correct or not (although, the model predictions for the sampled 4 cases are all correct).
Revised: • 3 are due to wrong “ground truth” – the given label is wrong regardless whether the prediction is correct or not (although, the model predictions for the sampled 4 cases are all correct).

Original: 8.5. E XAMPLE OF WRONG PROBLEM DECOMPOSITION
Revised: 8.5. E XAMPLE OF WRONG PROBLEM DECOMPOSITION

Original: In the following example, the decomposed question is just a rephrase of the original question.
Revised: In the following example, the decomposed question is just a rephrase of the original question.

Original: Q: Then, in 1544, ﬁve French galleys under Polin, including the superb R´eale, accompanied Barbarossa’s ﬂeet, on a diplomatic mission to Suleiman.
Revised: Q: Then, in 1544, ﬁve French galleys under Polin, including the superb R´eale, accompanied Barbarossa’s ﬂeet, on a diplomatic mission to Suleiman.

Original: The French ﬂeet accompanied Barbarossa during his attacks on the west coast of Italy on the way to Constantinople, as he laid waste to the cities of Porto Ercole, Giglio, Talamona, Lipari and took about 6,000 captives, but separated in
Revised: The French ﬂeet accompanied Barbarossa during his attacks on the west coast of Italy on the way to Constantinople, as he laid waste to the cities of Porto Ercole, Giglio, Talamona, Lipari and took about 6,000 captives, but separated in

Original: Sicily from Barbarossa’s ﬂeet to continue alone to the Ottoman capital.
Revised: Sicily from Barbarossa’s ﬂeet to continue alone to the Ottoman capital.

Original: Jerˆome Maurand, a priest of Antibes who accompanied Polin and the Ottoman ﬂeet in 1544, wrote a detailed account in
Revised: Jerˆome Maurand, a priest of Antibes who accompanied Polin and the Ottoman ﬂeet in 1544, wrote a detailed account in

Original: Itin´eraire d’Antibes `a Constantinople.
Revised: Itin´eraire d’Antibes `a Constantinople.

Original: They arrived in Constantinople on 10 August 1544 to meet with Suleiman and give him an account of the campaign.
Revised: They arrived in Constantinople on 10 August 1544 to meet with Suleiman and give him an account of the campaign.

Original: Polin was back to Toulon on 2 October 1544.
Revised: Polin was back to Toulon on 2 October 1544.

Original: On his way to Constantinople, how many cities did Polin laid waste to?
Revised: On his way to Constantinople, how many cities did Polin laid waste to?

Original: A: To answer the question ”On his way to Constantinople, how many cities did Polin laid waste to?”, we need to know: ”How many cities did Polin laid waste to on his way to Constantinople?”.
Revised: A: To answer the question ”On his way to Constantinople, how many cities did Polin laid waste to?”, we need to know: ”How many cities did Polin laid waste to on his way to Constantinople?”.

Original: 8.5. E XAMPLE OF WRONG PROBLEM SOLVING
Revised: 8.5. E XAMPLE OF WRONG PROBLEM SOLVING

Original: In the following example, the answer to the decomposed question, “How many percent of people were from 2 or more races in 2000?”
Revised: In the following example, the answer to the decomposed question, “How many percent of people were from 2 or more races in 2000?”

Original: is wrong.
Revised: is wrong.

Original: As of 2011 the 88,112 residents of Orem had a racial and ethnic composition of 89.3% White
Revised: As of 2011 the 88,112 residents of Orem had a racial and ethnic composition of 89.3% White

Original: American, 0.9% black or African American, 0.9% Native Americans of the United States, 1.6%
Revised: American, 0.9% black or African American, 0.9% Native Americans of the United States, 1.6%

Original: Asian American, 0.5% Paciﬁc Islanders, 4% non-Hispanics reporting some other race, 2.9% two or more races reported and 14.8% Hispanic, as Orem has a large Mexican American community with other Latinos residing in the city.
Revised: Asian American, 0.5% Paciﬁc Islanders, 4% non-Hispanics reporting some other race, 2.9% two or more races reported and 14.8% Hispanic, as Orem has a large Mexican American community with other Latinos residing in the city.

Original: This contrasts with the census of 2000, which showed a racial makeup of 90.80% White (U.S. Census), 0.33% African American (U.S. Census), 0.73%
Revised: This contrasts with the census of 2000, which showed a racial makeup of 90.80% White (U.S. Census), 0.33% African American (U.S. Census), 0.73%

Original: Native American (U.S. Census), 1.45% Asian (U.S. Census), 0.86% Paciﬁc Islander (U.S. Census), 3.64% from Race (United States Census), and 2.18% from two or more races.
Revised: Native American (U.S. Census), 1.45% Asian (U.S. Census), 0.86% Paciﬁc Islander (U.S. Census), 3.64% from Race (United States Census), and 2.18% from two or more races.

Original: Hispanic (U.S.
Revised: Hispanic (U.S.

Original: Census) or Latino (U.S. Census) of any race were 8.56% of the population.
Revised: Census) or Latino (U.S. Census) of any race were 8.56% of the population.

Original: The 2000 Census counted 84,324 people, 23,382 households, and 19,079 families.
Revised: The 2000 Census counted 84,324 people, 23,382 households, and 19,079 families.

Original: The population density at that time was 4,572.6 people per square mile (1,765.6/km 2 ).
Revised: The population density at that time was 4,572.6 people per square mile (1,765.6/km 2 ).

Original: There were 24,166 housing units at an average density of 1,310.4 per square mile (506.0/km 2 ).
Revised: There were 24,166 housing units at an average density of 1,310.4 per square mile (506.0/km 2 ).

Original: There were 23,382 households out of which 48.8% had children under the age of 18 living with them, 69.0% were Marriage living together, 9.5% had a female householder with no husband present, and 18.4% were non-families.
Revised: There were 23,382 households out of which 48.8% had children under the age of 18 living with them, 69.0% were Marriage living together, 9.5% had a female householder with no husband present, and 18.4% were non-families.

Original: 12.4% of all households were made up of individuals and 5.1% had someone living alone who was 65 years of age or older.
Revised: 12.4% of all households were made up of individuals and 5.1% had someone living alone who was 65 years of age or older.

Original: The average household size was 3.57 and the average family size was 3.93.
Revised: The average household size was 3.57 and the average family size was 3.93.

Original: Q: How many percent of people were from 2 or more races in 2000?
Revised: Q: How many percent of people were from 2 or more races in 2000?

Original: A: “3.64% from Race (United States Census), and 2.18% from two or more races”.
Revised: A: “3.64% from Race (United States Census), and 2.18% from two or more races”.

Original: So the answer is 3.64% + 2.18% = 5.82%.
Revised: So the answer is 3.64% + 2.18% = 5.82%.

Original: Q: How many percent of people were not from 2 or more races in 2000?
Revised: Q: How many percent of people were not from 2 or more races in 2000?

Original: A:
Revised: A:

Original: We know that 5.82% were from 2 or more races.
Revised: We know that 5.82% were from 2 or more races.

Original: So 100% - 5.82% = 94.18% were not from 2 or more races.
Revised: So 100% - 5.82% = 94.18% were not from 2 or more races.

Original: So the answer is 94.18%.
Revised: So the answer is 94.18%.

Original: 8.5. E XAMPLE OF WRONG GIVEN LABEL
Revised: 8.5. E XAMPLE OF WRONG GIVEN LABEL

Original: In the example below, the given label is 417, which is incorrect.
Revised: In the example below, the given label is 417, which is incorrect.

Original: The correct ground truth should be 418.
Revised: The correct ground truth should be 418.

Original: The model gives the correct answer via least-to-most prompting.
Revised: The model gives the correct answer via least-to-most prompting.

Original: There were 20,928 births in 2006.
Revised: There were 20,928 births in 2006.

Original: Of these, 19,757 (94.40% of the births, 95.19% of the population) were to Non-Hispanic Whites.
Revised: Of these, 19,757 (94.40% of the births, 95.19% of the population) were to Non-Hispanic Whites.

Original: There were 22 births to American Indians (0.11% of the births and 0.54% of the population), 177 births to Asians (0.85% of the births and 0.68% of the population), 219 births to Hispanics (1.05% of the births and 0.88% of the population) and 753 births to Blacks and others (3.60% of the births and 3.56% of the population).
Revised: There were 22 births to American Indians (0.11% of the births and 0.54% of the population), 177 births to Asians (0.85% of the births and 0.68% of the population), 219 births to Hispanics (1.05% of the births and 0.88% of the population) and 753 births to Blacks and others (3.60% of the births and 3.56% of the population).

Original: Q: How many births were there for Asians?
Revised: Q: How many births were there for Asians?

Original: A: “177 births to Asians”.
Revised: A: “177 births to Asians”.

Original: So the answer is 177.
Revised: So the answer is 177.

Original: Q
Revised: Q

Original: : How many births were there for Native Americans?
Revised: : How many births were there for Native Americans?

Original: A: “22 births to American Indians”.
Revised: A: “22 births to American Indians”.

Original: So the answer is 22.
Revised: So the answer is 22.

Original: Q: How many births were there for Hispanics?
Revised: Q: How many births were there for Hispanics?

Original: A: “219 births to Hispanics”.
Revised: A: “219 births to Hispanics”.

Original: So the answer is 219.
Revised: So the answer is 219.

Original: Q: How many total births were there for Asians, Native Americans and Hispanics?
Revised: Q: How many total births were there for Asians, Native Americans and Hispanics?

Original: A: We know that there were 177 births for Asians.
Revised: A: We know that there were 177 births for Asians.

Original: We also know that there were 22 births for Native
Revised: We also know that there were 22 births for Native

Original: Americans.
Revised: Americans.

Original: We also know that there were 219 births for Hispanics.
Revised: We also know that there were 219 births for Hispanics.

Original: So 177 + 22 + 219 = 418 total births for Asians, Native Americans and Hispanics.
Revised: So 177 + 22 + 219 = 418 total births for Asians, Native Americans and Hispanics.

Original: So the answer is 418.
Revised: So the answer is 418.

Original:  GSM8K 9. E XPERIMENT RESULTS : O NE - SHOT PROMPTS
Revised:  GSM8K 9. E XPERIMENT RESULTS : O NE - SHOT PROMPTS

Original: We compare here the effectiveness on compositional generalization of least-to-most prompting vs. chain-of-thought prompting by constructing for each prompting method a simple prompt context that contains a single example that is solvable with just 2 reasoning steps.
Revised: We compare here the effectiveness on compositional generalization of least-to-most prompting vs. chain-of-thought prompting by constructing for each prompting method a simple prompt context that contains a single example that is solvable with just 2 reasoning steps.

Original: We then evaluate accuracy on examples that may involve larger numbers of reasoning steps.
Revised: We then evaluate accuracy on examples that may involve larger numbers of reasoning steps.

Original: The same example is used for both prompting methods.
Revised: The same example is used for both prompting methods.

Original: For the least-to-most prompting prompt, we adopt a simpliﬁed approach in which the problem decomposition and solution stages are merged into a single pass, with just one follow-up request being made to the language model to solicit the ﬁnal answer.
Revised: For the least-to-most prompting prompt, we adopt a simpliﬁed approach in which the problem decomposition and solution stages are merged into a single pass, with just one follow-up request being made to the language model to solicit the ﬁnal answer.

Original: The accuracy (%) of the two prompting methods with the GPT-3 code-davinci-002 model, with breakdown by number of steps in the expected solution, are listed in Table 18.
Revised: The accuracy (%) of the two prompting methods with the GPT-3 code-davinci-002 model, with breakdown by number of steps in the expected solution, are listed in Table 18.

Original: Accuracy numbers for all prompting methods are calculated after applying the same post-processing as described in Section 3.3 for DROP.
Revised: Accuracy numbers for all prompting methods are calculated after applying the same post-processing as described in Section 3.3 for DROP.

Original: While the least-to-most prompting accuracy is overall only moderately higher than that of chain-ofthought prompting, the accuracy breakdown by number of steps shows that least-to-most prompting signiﬁcantly outperforms chain-of-thought prompting as the number of reasoning steps increases beyond what was illustrated in the prompt.
Revised: While the least-to-most prompting accuracy is overall only moderately higher than that of chain-ofthought prompting, the accuracy breakdown by number of steps shows that least-to-most prompting signiﬁcantly outperforms chain-of-thought prompting as the number of reasoning steps increases beyond what was illustrated in the prompt.

Original: Accuracy by Steps
Revised: Accuracy by Steps

Original: All   5+
Revised: All   5+

Original: Least-to-
Revised: Least-to-

Original: [Table]
Revised: [Table]

Original: 9. E XPERIMENT RESULTS : E NGINEERED PROMPTS
Revised: 9. E XPERIMENT RESULTS : E NGINEERED PROMPTS

Original: We compare here the overall accuracy of the above-reported “Chain-of-Thought (1-shot)” and “Least-to-Most (1-shot)” methods with alternative existing prompting methods, as well as with variants of chain-of-thought and least-to-most prompting in which the prompts were engineered using multiple in-domain examples taken from the GSM8K train set.
Revised: We compare here the overall accuracy of the above-reported “Chain-of-Thought (1-shot)” and “Least-to-Most (1-shot)” methods with alternative existing prompting methods, as well as with variants of chain-of-thought and least-to-most prompting in which the prompts were engineered using multiple in-domain examples taken from the GSM8K train set.

Original: The evaluated prompting methods are as follows (see Appendices 9.3 and 9.4 for the exact prompt contexts):
Revised: The evaluated prompting methods are as follows (see Appendices 9.3 and 9.4 for the exact prompt contexts):

Original: • Zero-Shot : Simple zero-shot prompting.
Revised: • Zero-Shot : Simple zero-shot prompting.

Original: • Standard prompting : Standard few-shot prompting, using the same 4 examples as in the “problem solving” prompt context of “Least-to-Most (best)”.
Revised: • Standard prompting : Standard few-shot prompting, using the same 4 examples as in the “problem solving” prompt context of “Least-to-Most (best)”.

Original: • Chain-of-Thought (original) : Chain-of-thought prompting, using the original 8-shot prompt context described in Wei et al.
Revised: • Chain-of-Thought (original) : Chain-of-thought prompting, using the original 8-shot prompt context described in Wei et al.

Original: • Chain-of-Thought (1-shot) :
Revised: • Chain-of-Thought (1-shot) :

Original: The simple 1-shot chain-of-thought prompting method described in Appendix 9.1 above.
Revised: The simple 1-shot chain-of-thought prompting method described in Appendix 9.1 above.

Original: • Least-to-Most (1-shot) :
Revised: • Least-to-Most (1-shot) :

Original: The simple 1-shot least-to-most prompting method described in Appendix 9.1 above.
Revised: The simple 1-shot least-to-most prompting method described in Appendix 9.1 above.

Original: Chain-of-Thought (best) :
Revised: Chain-of-Thought (best) :

Original: Chain-of-thought prompting, using the same 4 examples as in the “problem solving” prompt context of “Least-to-Most (best)”, with the solutions adjusted to chain-of-thought format.
Revised: Chain-of-thought prompting, using the same 4 examples as in the “problem solving” prompt context of “Least-to-Most (best)”, with the solutions adjusted to chain-of-thought format.

Original: • Least-to-Most (best) : Least-to-most prompting using separate prompts for the “problem decomposition” and “problem solution” steps and with multiple examples selected from the GSM8K train set.
Revised: • Least-to-Most (best) : Least-to-most prompting using separate prompts for the “problem decomposition” and “problem solution” steps and with multiple examples selected from the GSM8K train set.

Original: The “problem decomposition” prompt context contains 7 examples, with hand-crafted problem decompositions.
Revised: The “problem decomposition” prompt context contains 7 examples, with hand-crafted problem decompositions.

Original: The “problem solution” prompt containsexamples, with hand-crafted solutions for each step.
Revised: The “problem solution” prompt containsexamples, with hand-crafted solutions for each step.

Original: The accuracies (%) of these prompting methods with the GPT-3 code-davinci-002 model are listed in Table 19.
Revised: The accuracies (%) of these prompting methods with the GPT-3 code-davinci-002 model are listed in Table 19.

Original: It can be noted ﬁrst that, although the “Chain-of-Thought (1-shot)” prompt context is considerably simpler than the 8-shot prompt context proposed in the original chain-of-thought paper, we ﬁnd the overall accuracy achieved to be quite close (60.87% for the 1-shot prompt, compared to 61.18% for the original 8-shot prompt).
Revised: It can be noted ﬁrst that, although the “Chain-of-Thought (1-shot)” prompt context is considerably simpler than the 8-shot prompt context proposed in the original chain-of-thought paper, we ﬁnd the overall accuracy achieved to be quite close (60.87% for the 1-shot prompt, compared to 61.18% for the original 8-shot prompt).

Original: This suggests that “Chain-of-Thought (1-shot)” is indeed a reasonable chain-of-thought baseline to analyze in comparison to “Least-to-Most (1-shot)”.
Revised: This suggests that “Chain-of-Thought (1-shot)” is indeed a reasonable chain-of-thought baseline to analyze in comparison to “Least-to-Most (1-shot)”.

Original: Further, while we ﬁnd the proposed 1-shot prompt attractive due to its simplicity and lack of datasetspeciﬁc content, we do ﬁnd that further improvements in overall accuracy of both chain-of-thought and least-to-most prompting can be achieved if additional prompt engineering is applied, using multiple in-domain examples of arbitrary complexity from the GSM8K train set, as seen in the accuracies of “Chain-of-Thought (best)” and “Least-to-Most (best)”.
Revised: Further, while we ﬁnd the proposed 1-shot prompt attractive due to its simplicity and lack of datasetspeciﬁc content, we do ﬁnd that further improvements in overall accuracy of both chain-of-thought and least-to-most prompting can be achieved if additional prompt engineering is applied, using multiple in-domain examples of arbitrary complexity from the GSM8K train set, as seen in the accuracies of “Chain-of-Thought (best)” and “Least-to-Most (best)”.

Original: We do not observe improvement in overall accuracy from least-to-most prompting compared to chain-of-thought prompting in this setting, where most of the test questions do not require more steps more steps to solve than the demonstration examples.
Revised: We do not observe improvement in overall accuracy from least-to-most prompting compared to chain-of-thought prompting in this setting, where most of the test questions do not require more steps more steps to solve than the demonstration examples.

Original: [Table]
Revised: [Table]

Original: 9. P ROMPT CONTEXTS : O NE - SHOT PROMPTS
Revised: 9. P ROMPT CONTEXTS : O NE - SHOT PROMPTS

Original: We include here the prompt contexts used in the experiments reported in Appendix 9.1.
Revised: We include here the prompt contexts used in the experiments reported in Appendix 9.1.

Original: In this section and in the following one, the placeholder “ { question } ” indicates the place where the original question is to be inserted, in cases where the format would not be obvious (e.g., where instead of simply ending the prompt with “A:”, we include some additional prompt text like “A:
Revised: In this section and in the following one, the placeholder “ { question } ” indicates the place where the original question is to be inserted, in cases where the format would not be obvious (e.g., where instead of simply ending the prompt with “A:”, we include some additional prompt text like “A:

Original: The answer is”).
Revised: The answer is”).

Original: In the case of “Least-to-Most (1-shot)”, the prompt preﬁx for the initial request ends in “Let’s break down this problem:”.
Revised: In the case of “Least-to-Most (1-shot)”, the prompt preﬁx for the initial request ends in “Let’s break down this problem:”.

Original: We then append to that prompt the initial reply that was received from the language model, followed by a newline and the string “The answer is:”, which we then use as the prompt in a second request, whose reply we treat as the ﬁnal answer.
Revised: We then append to that prompt the initial reply that was received from the language model, followed by a newline and the string “The answer is:”, which we then use as the prompt in a second request, whose reply we treat as the ﬁnal answer.

Original: 9.3. C HAIN - OF -T HOUGHT (1- SHOT )
Revised: 9.3. C HAIN - OF -T HOUGHT (1- SHOT )

Original: Q: Elsa has 5 apples.
Revised: Q: Elsa has 5 apples.

Original: Anna has 2 more apples than Elsa.
Revised: Anna has 2 more apples than Elsa.

Original: How many apples do they have together?
Revised: How many apples do they have together?

Original: A: Anna has 2 more apples than Elsa, so Anna has 2 + 5 = 7 apples.
Revised: A: Anna has 2 more apples than Elsa, so Anna has 2 + 5 = 7 apples.

Original: Elsa and Anna have 5 + 7 =apples together.
Revised: Elsa and Anna have 5 + 7 =apples together.

Original: The answer is 12.
Revised: The answer is 12.

Original: 9.3. L EAST - TO -M OST (1- SHOT )
Revised: 9.3. L EAST - TO -M OST (1- SHOT )

Original: Q: Elsa has 5 apples.
Revised: Q: Elsa has 5 apples.

Original: Anna has 2 more apples than Elsa.
Revised: Anna has 2 more apples than Elsa.

Original: How many apples do they have together?
Revised: How many apples do they have together?

Original: A: Let’s break down this problem: 1.
Revised: A: Let’s break down this problem: 1.

Original: How many apples does Anna have? 2.
Revised: How many apples does Anna have? 2.

Original: How many apples do Elsa and Anna have together?
Revised: How many apples do Elsa and Anna have together?

Original: Anna has 2 more apples than Elsa.
Revised: Anna has 2 more apples than Elsa.

Original: So Anna has 2 + 5 = 7 apples.
Revised: So Anna has 2 + 5 = 7 apples.

Original: Elsa and Anna have 5 + 7 = 12 apples together.
Revised: Elsa and Anna have 5 + 7 = 12 apples together.

Original: Q: { question } A: Let’s break down this problem: —–
Revised: Q: { question } A: Let’s break down this problem: —–

Original: The answer is:
Revised: The answer is:

Original: 9. P ROMPT CONTEXTS : E NGINEERED PROMPTS
Revised: 9. P ROMPT CONTEXTS : E NGINEERED PROMPTS

Original: We include here the additional prompt templates used in the experiments reported in Appendix 9.2, with the exception of “Chain-of-Thought (original)“, which is the same as the “Prompt for Math
Revised: We include here the additional prompt templates used in the experiments reported in Appendix 9.2, with the exception of “Chain-of-Thought (original)“, which is the same as the “Prompt for Math

Original: Word Problems” reported in Appendix G of Wei et al.
Revised: Word Problems” reported in Appendix G of Wei et al.

Original: 9.4. Z ERO -S HOT Q: { question }
Revised: 9.4. Z ERO -S HOT Q: { question }

Original: A
Revised: A

Original: The answer is
Revised: The answer is

Original: 9.4. S TANDARD PROMPTING : 4 EXAMPLES
Revised: 9.4. S TANDARD PROMPTING : 4 EXAMPLES

Original: Q: Four years ago, Kody was only half as old as Mohamed.
Revised: Q: Four years ago, Kody was only half as old as Mohamed.

Original: If Mohamed is currently twice 30 years old, how old is
Revised: If Mohamed is currently twice 30 years old, how old is

Original: Kody?
Revised: Kody?

Original: A:
Revised: A:

Original: The answer is 32.
Revised: The answer is 32.

Original: Q: Carla bought 2 bags of mini peanut butter cups on clearance.
Revised: Q: Carla bought 2 bags of mini peanut butter cups on clearance.

Original: Each bag was $6.00 but was 75% off.
Revised: Each bag was $6.00 but was 75% off.

Original: How much did she spend on 2 bags of candy?
Revised: How much did she spend on 2 bags of candy?

Original: A:
Revised: A:

Original: The answer is 3.
Revised: The answer is 3.

Original: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?
Revised: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?

Original: A:
Revised: A:

Original: The answer is 5.
Revised: The answer is 5.

Original: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.
Revised: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.

Original: Sandy orders some drinks for herself and some friends.
Revised: Sandy orders some drinks for herself and some friends.

Original: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.
Revised: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.

Original: How much change does she receive back for a twenty-dollar bill?
Revised: How much change does she receive back for a twenty-dollar bill?

Original: A:
Revised: A:

Original: The answer is 3.
Revised: The answer is 3.

Original: 9.4. C HAIN - OF -T HOUGHT ( BEST ): 4 EXAMPLES
Revised: 9.4. C HAIN - OF -T HOUGHT ( BEST ): 4 EXAMPLES

Original: Q
Revised: Q

Original: : Four years ago, Kody was only half as old as Mohamed.
Revised: : Four years ago, Kody was only half as old as Mohamed.

Original: If Mohamed is currently twice 30 years old, how old is Kody?
Revised: If Mohamed is currently twice 30 years old, how old is Kody?

Original: A:
Revised: A:

Original: We were told that Mohamed is currently twice 30 years old, so he is currently 30 * 2 = 60 years old.
Revised: We were told that Mohamed is currently twice 30 years old, so he is currently 30 * 2 = 60 years old.

Original: That means that four years ago he must have been 60 - 4 = 56 years old.
Revised: That means that four years ago he must have been 60 - 4 = 56 years old.

Original: Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 = 28 years old then.
Revised: Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 = 28 years old then.

Original: Since Kody was 28 years old four years ago, she must now be 28 + 4 = 32 years old.
Revised: Since Kody was 28 years old four years ago, she must now be 28 + 4 = 32 years old.

Original: The answer is 32.
Revised: The answer is 32.

Original: Q: Carla bought 2 bags of mini peanut butter cups on clearance.
Revised: Q: Carla bought 2 bags of mini peanut butter cups on clearance.

Original: Each bag was $6.00 but was 75% off.
Revised: Each bag was $6.00 but was 75% off.

Original: How much did she spend on 2 bags of candy?
Revised: How much did she spend on 2 bags of candy?

Original: A: Each bag was $6.00 but was 75% off.
Revised: A: Each bag was $6.00 but was 75% off.

Original: So each bag cost $6.00 * (1 - 0.75) =
Revised: So each bag cost $6.00 * (1 - 0.75) =

Original: $6.00 * 0.25 = $1.50.
Revised: $6.00 * 0.25 = $1.50.

Original: Carla bought 2 bags.
Revised: Carla bought 2 bags.

Original: So she spent $1.50 * 2 = $3.00.
Revised: So she spent $1.50 * 2 = $3.00.

Original: The answer is 3.
Revised: The answer is 3.

Original: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?
Revised: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?

Original: A: Since Rena will be
Revised: A: Since Rena will be

Original: years
Revised: years

Original: older than
Revised: older than

Original: Pam
Revised: Pam

Original: in
Revised: in

Original: 10 years
Revised: 10 years

Original: , she must be 5 years older than Pam now
Revised: , she must be 5 years older than Pam now

Original: as well
Revised: as well

Original: If
Revised: If

Original: Pam
Revised: Pam

Original: is currently twice as young as Rena,
Revised: is currently twice as young as Rena,

Original: that means that
Revised: that means that

Original: Rena is currently twice
Revised: Rena is currently twice

Original: as old
Revised: as old

Original: as
Revised: as

Original: Pam
Revised: Pam

Original: is.
Revised: is.

Original: So if P stands for Pam’s age now and R stands for Rena’s age now, then we know that R = 2 * P
Revised: So if P stands for Pam’s age now and R stands for Rena’s age now, then we know that R = 2 * P

Original: And
Revised: And

Original: since Rena is 5 years older than Pam now, we know that R = P + 5.
Revised: since Rena is 5 years older than Pam now, we know that R = P + 5.

Original: By substitution, we have P + 5 = 2 * P,
Revised: By substitution, we have P + 5 = 2 * P,

Original: which
Revised: which

Original: means that P = 5.
Revised: means that P = 5.

Original: The answer is 5.
Revised: The answer is 5.

Original: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and
Revised: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and

Original: espressos
Revised: espressos

Original: cost $1 each.
Revised: cost $1 each.

Original: Sandy orders some drinks for herself and some friends.
Revised: Sandy orders some drinks for herself and some friends.

Original: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.
Revised: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.

Original: How much change does she receive back for a twenty-dollar bill?
Revised: How much change does she receive back for a twenty-dollar bill?

Original: A: Sandy ordered three cappuccinos, which cost $2 each, so she spent $2 * 3 = $6 on cappuccinos.
Revised: A: Sandy ordered three cappuccinos, which cost $2 each, so she spent $2 * 3 = $6 on cappuccinos.

Original: She ordered two iced teas, which cost $3 each, so she spent $3 * 2 = $6 dollars on ice teas.
Revised: She ordered two iced teas, which cost $3 each, so she spent $3 * 2 = $6 dollars on ice teas.

Original: She ordered two cafe lattes, which cost $1.5 each, so she spent $1.5 * 2 = $3 on cafe lattes.
Revised: She ordered two cafe lattes, which cost $1.5 each, so she spent $1.5 * 2 = $3 on cafe lattes.

Original: She ordered two espressos, which cost $1 each, so she spent $1 * 2 = $2 on espressos.
Revised: She ordered two espressos, which cost $1 each, so she spent $1 * 2 = $2 on espressos.

Original: So altogether, Sandy spent $6 + $6 + $3 + $2 = $17 on drinks, which means that sandy will get $20 - $17 = $3 as change.
Revised: So altogether, Sandy spent $6 + $6 + $3 + $2 = $17 on drinks, which means that sandy will get $20 - $17 = $3 as change.

Original: The answer is 3.
Revised: The answer is 3.

Original: 9.4. L EAST - TO -M OST ( BEST ) I - PROBLEM DECOMPOSITION : 7 EXAMPLES
Revised: 9.4. L EAST - TO -M OST ( BEST ) I - PROBLEM DECOMPOSITION : 7 EXAMPLES

Original: Q: Four years ago, Kody was only half as old as Mohamed.
Revised: Q: Four years ago, Kody was only half as old as Mohamed.

Original: If Mohamed is currently twice 30 years old, how old is Kody?
Revised: If Mohamed is currently twice 30 years old, how old is Kody?

Original: A: To answer the question “How old is Kody?”
Revised: A: To answer the question “How old is Kody?”

Original: , we need to know: “How old is Mohamed?”, “How old was Mohamed four years ago?”, “How old was Kody four years ago?”.
Revised: , we need to know: “How old is Mohamed?”, “How old was Mohamed four years ago?”, “How old was Kody four years ago?”.

Original: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?
Revised: Q: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?

Original: A: To answer the question “How old is Pam now?”
Revised: A: To answer the question “How old is Pam now?”

Original: , we need to know: “How much older is Rena than Pam currently?”.
Revised: , we need to know: “How much older is Rena than Pam currently?”.

Original: Q: As a freelancer, Baylor is paid for every ﬁnished work of a client he does on a freelance marketplace.
Revised: Q: As a freelancer, Baylor is paid for every ﬁnished work of a client he does on a freelance marketplace.

Original: Currently, he has $4000 on his dashboard from previous work done.
Revised: Currently, he has $4000 on his dashboard from previous work done.

Original: He is currently working for three clients, with the ﬁrst client paying him half the amount of money he currently has on his dashboard once the job is done.
Revised: He is currently working for three clients, with the ﬁrst client paying him half the amount of money he currently has on his dashboard once the job is done.

Original: The second client will pay him 2/5 times more money than the ﬁrst client once Baylor ﬁnishes his work.
Revised: The second client will pay him 2/5 times more money than the ﬁrst client once Baylor ﬁnishes his work.

Original: The third client will pay him twice the amount of money the ﬁrst and second clients pay him together once he ﬁnishes the
Revised: The third client will pay him twice the amount of money the ﬁrst and second clients pay him together once he ﬁnishes the

Original: job.
Revised: job.

Original: How much money will Baylor have in his dashboard after all the clients pay him for his work?
Revised: How much money will Baylor have in his dashboard after all the clients pay him for his work?

Original: A:
Revised: A:

Original: To answer the question “How much money will Baylor have in his dashboard after all the clients pay him for his work?”
Revised: To answer the question “How much money will Baylor have in his dashboard after all the clients pay him for his work?”

Original: , we need to know: “How much will Baylor’s ﬁrst client pay him for his work?”, “How much more will Baylor’s second client pay him for his work compared to the ﬁrst client?”, “How much will Baylor’s second client pay him for his work?”, “How much will the ﬁrst and second clients pay him together once he ﬁnishes the job?”, “How much will Baylor’s third client pay him for his work?”, “How much money will all the clients pay Baylor for his work?”.
Revised: , we need to know: “How much will Baylor’s ﬁrst client pay him for his work?”, “How much more will Baylor’s second client pay him for his work compared to the ﬁrst client?”, “How much will Baylor’s second client pay him for his work?”, “How much will the ﬁrst and second clients pay him together once he ﬁnishes the job?”, “How much will Baylor’s third client pay him for his work?”, “How much money will all the clients pay Baylor for his work?”.

Original: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.
Revised: Q: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.

Original: Sandy orders some drinks for herself and some friends.
Revised: Sandy orders some drinks for herself and some friends.

Original: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.
Revised: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.

Original: How much change does she receive back for a twenty-dollar bill?
Revised: How much change does she receive back for a twenty-dollar bill?

Original: A:To answer the question “How much change does she receive back for a twenty-dollar bill?”
Revised: A:To answer the question “How much change does she receive back for a twenty-dollar bill?”

Original: , we need to know: “How much did the cappuccinos cost in total?”, “How much did the iced teas cost in total?”, “How much did the cafe lattes cost in total?”, “How much did the espressos cost in total?”, “How much did the drinks cost in total?”.
Revised: , we need to know: “How much did the cappuccinos cost in total?”, “How much did the iced teas cost in total?”, “How much did the cafe lattes cost in total?”, “How much did the espressos cost in total?”, “How much did the drinks cost in total?”.

Original: Q: Betty & Paige are raising money for their kids’ little league team by hosting a bake sale.
Revised: Q: Betty & Paige are raising money for their kids’ little league team by hosting a bake sale.

Original: Betty has baked 4 dozen chocolate chip cookies, 6 dozen oatmeal raisin cookies and 2 dozen regular brownies.
Revised: Betty has baked 4 dozen chocolate chip cookies, 6 dozen oatmeal raisin cookies and 2 dozen regular brownies.

Original: Paige baked 6 dozen sugar cookies, 3 dozen blondies and 5 dozen cream cheese swirled brownies.
Revised: Paige baked 6 dozen sugar cookies, 3 dozen blondies and 5 dozen cream cheese swirled brownies.

Original: If they sell the cookies for $1.00 apiece and the blondies/brownies at $2.00 apiece, how much money will they raise?
Revised: If they sell the cookies for $1.00 apiece and the blondies/brownies at $2.00 apiece, how much money will they raise?

Original: A: To answer the question “How much money will they raise?”, we need to know: “How many dozen cookies did they bake (not including blondies/brownies)?”
Revised: A: To answer the question “How much money will they raise?”, we need to know: “How many dozen cookies did they bake (not including blondies/brownies)?”

Original: , “How many cookies did they bake (not including blondies/brownies)?”, “How many dozen blondies/brownies did they bake (not including cookies)?”, “How many blondies/brownies did they bake (not including cookies)?”, “How much money will they raise from the cookies (not including blondies/brownies)?”, “How much money will they raise from the blondies/brownies (not including cookies)?”.
Revised: , “How many cookies did they bake (not including blondies/brownies)?”, “How many dozen blondies/brownies did they bake (not including cookies)?”, “How many blondies/brownies did they bake (not including cookies)?”, “How much money will they raise from the cookies (not including blondies/brownies)?”, “How much money will they raise from the blondies/brownies (not including cookies)?”.

Original: Q: On a moonless night, three ﬁreﬂies danced in the evening breeze.
Revised: Q: On a moonless night, three ﬁreﬂies danced in the evening breeze.

Original: They were joined by four less than a dozen more ﬁreﬂies, before two of the ﬁreﬂies ﬂew away.
Revised: They were joined by four less than a dozen more ﬁreﬂies, before two of the ﬁreﬂies ﬂew away.

Original: How many ﬁreﬂies remained?
Revised: How many ﬁreﬂies remained?

Original: A: To answer the question “How many ﬁreﬂies remained?”
Revised: A: To answer the question “How many ﬁreﬂies remained?”

Original: , we need to know: “How many ﬁreﬂies joined?”.
Revised: , we need to know: “How many ﬁreﬂies joined?”.

Original: Q: Sam, Sid, and Steve brought popsicle sticks for their group activity in their Art class.
Revised: Q: Sam, Sid, and Steve brought popsicle sticks for their group activity in their Art class.

Original: Sam has thrice as many as Sid, and Sid has twice as many as Steve.
Revised: Sam has thrice as many as Sid, and Sid has twice as many as Steve.

Original: If Steve has 12 popsicle sticks, how many popsicle sticks can they use for their Art class activity?
Revised: If Steve has 12 popsicle sticks, how many popsicle sticks can they use for their Art class activity?

Original: A: To answer the question “How many popsicle sticks can they use for their Art class activity?”, we need to know: “How many popsicle sticks does Sid have?”, “How many popsicle sticks does Sam have?”.
Revised: A: To answer the question “How many popsicle sticks can they use for their Art class activity?”, we need to know: “How many popsicle sticks does Sid have?”, “How many popsicle sticks does Sam have?”.

Original: 9.4. L EAST - TO -M OST ( BEST ) II - PROBLEM SOLVING : 4 EXAMPLES
Revised: 9.4. L EAST - TO -M OST ( BEST ) II - PROBLEM SOLVING : 4 EXAMPLES

Original: Four years ago, Kody was only half as old as Mohamed.
Revised: Four years ago, Kody was only half as old as Mohamed.

Original: If Mohamed is currently twice 30 years
Revised: If Mohamed is currently twice 30 years

Original: old
Revised: old

Original: , how old is
Revised: , how old is

Original: Kody
Revised: Kody

Original: Q: How old was Mohamed four years ago?
Revised: Q: How old was Mohamed four years ago?

Original: A:
Revised: A:

Original: We were told that Mohamed is currently twice 30 years old, so he is currently 30 * 2 = 60 years old.
Revised: We were told that Mohamed is currently twice 30 years old, so he is currently 30 * 2 = 60 years old.

Original: That means that four years ago he must have been 60 - 4 = 56 years old.
Revised: That means that four years ago he must have been 60 - 4 = 56 years old.

Original: The answer is 56.
Revised: The answer is 56.

Original: Q: How old is Kody?
Revised: Q: How old is Kody?

Original: A:
Revised: A:

Original: Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 = 28 years old then.
Revised: Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 = 28 years old then.

Original: Since Kody was 28 years old four years ago, she must now be 28 + 4 = 32 years old.
Revised: Since Kody was 28 years old four years ago, she must now be 28 + 4 = 32 years old.

Original: The answer is 32.
Revised: The answer is 32.

Original: Carla bought 2 bags of mini peanut butter cups on clearance.
Revised: Carla bought 2 bags of mini peanut butter cups on clearance.

Original: Each bag was $6.00 but was 75% off.
Revised: Each bag was $6.00 but was 75% off.

Original: How much did she spend on 2 bags of candy?
Revised: How much did she spend on 2 bags of candy?

Original: Q: How much did she spend on 2 bags of candy?
Revised: Q: How much did she spend on 2 bags of candy?

Original: A: Each bag was $6.00 but was 75% off.
Revised: A: Each bag was $6.00 but was 75% off.

Original: So each bag cost $6.00 * (1 - 0.75) =
Revised: So each bag cost $6.00 * (1 - 0.75) =

Original: $6.00 * 0.25 = $1.50.
Revised: $6.00 * 0.25 = $1.50.

Original: Carla bought 2 bags.
Revised: Carla bought 2 bags.

Original: So she spent $1.50 * 2 = $3.00.
Revised: So she spent $1.50 * 2 = $3.00.

Original: The answer is 3.
Revised: The answer is 3.

Original: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?
Revised: If Pam is currently twice as young as Rena is, and in 10 years Rena will be 5 years older than her, how old is Pam now?

Original: Q: How much older is Rena than Pam currently?
Revised: Q: How much older is Rena than Pam currently?

Original: A: Since Rena will be 5 years older than Pam in 10 years, she must be 5 years older than Pam now as well.
Revised: A: Since Rena will be 5 years older than Pam in 10 years, she must be 5 years older than Pam now as well.

Original: The answer is 5.
Revised: The answer is 5.

Original: Q: How old is Pam now?
Revised: Q: How old is Pam now?

Original: A: If Pam is currently twice as young as Rena, that means that Rena is currently twice as old as
Revised: A: If Pam is currently twice as young as Rena, that means that Rena is currently twice as old as

Original: Pam is.
Revised: Pam is.

Original: So if P stands for Pam’s age now and R stands for Rena’s age now, then we know that R = 

Revised: So if P stands for Pam’s age now and R stands for Rena’s age now, then we know that R =

Original: * P
Revised: * P

Original: And since Rena is 5 years older than Pam now, we know that R = P + 5.
Revised: And since Rena is 5 years older than Pam now, we know that R = P + 5.

Original: By substitution, we have P + 5 = 2 * P, which means that P = 5.
Revised: By substitution, we have P + 5 = 2 * P, which means that P = 5.

Original: The answer is 5.
Revised: The answer is 5.

Original: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.
Revised: Cappuccinos cost $2, iced teas cost $3, cafe lattes cost $1.5 and espressos cost $1 each.

Original: Sandy orders some drinks for herself and some friends.
Revised: Sandy orders some drinks for herself and some friends.

Original: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.
Revised: She orders three cappuccinos, two iced teas, two cafe lattes, and two espressos.

Original: How much change does she receive back for a twenty-dollar bill?
Revised: How much change does she receive back for a twenty-dollar bill?

Original: Q: How much did Sandy spend on drinks?
Revised: Q: How much did Sandy spend on drinks?

Original: A: Sandy ordered three cappuccinos, which cost $2 each, so she spent $2 * 3 = $6 on cappuccinos.
Revised: A: Sandy ordered three cappuccinos, which cost $2 each, so she spent $2 * 3 = $6 on cappuccinos.

Original: She ordered two iced teas, which cost $3 each, so she spent $3 * 2 = $6 dollars on ice teas.
Revised: She ordered two iced teas, which cost $3 each, so she spent $3 * 2 = $6 dollars on ice teas.

Original: She ordered two cafe lattes, which cost $1.5 each, so she spent $1.5 * 2 = $3 on cafe lattes.
Revised: She ordered two cafe lattes, which cost $1.5 each, so she spent $1.5 * 2 = $3 on cafe lattes.

Original: She ordered two espressos, which cost $1 each, so she spent $1 * 2 = $2 on espressos.
Revised: She ordered two espressos, which cost $1 each, so she spent $1 * 2 = $2 on espressos.

Original: So altogether, Sandy spent $6 + $6 + $3 + $2 = $17 on drinks.
Revised: So altogether, Sandy spent $6 + $6 + $3 + $2 = $17 on drinks.

Original: The answer is 17.
Revised: The answer is 17.