Questions on Word Problems: Numbers, consecutive odd/even, digits answered by real tutors! Questions on Word Problems: Numbers, consecutive odd/even, digits answered by real tutors! Tutors Answer Your Questions about Numbers Word Problems (FREE) Get help from our free tutors ===> You can put this solution on YOUR website! . x + y = 83, x - y = 15. Add the two equations. It is the standard method of solving problems like this. Question 1064026 : If you subtract 6 from a certain number, the difference is greater than four times the original number. For what numbers is this true? , subtracting six from a certain number, the difference... ,... is greater than four times the original number. Question 1063894 : The sum of two numbers is 71. One number is 25 more than the other. Find the numbers. x - y = 25 ------------------ Add Question 1063776 : The whole is equal to the sum of the parts. The sum of two consecutive number is 84 and one of them is more than others. What are the numbers? This problem does not have a solution.

The sum of two consecutive integers is an odd number. Always. 84 is not an odd number. Question 1063778 : equation(the sum of two consecutive integer is 145 ï¿½ and if you wanted to find x... 2x = 144 x = 72 ï¿½ so the numbers are 72 and 73 ... Question 1063733 : the sum of a number and 9 is multiplied by -2 and the answer is -8 find the number Found 2 solutions by amarjeeth123, ikleyn: Let the number be x. (x+9)(-2)=-8 You can put this solution on YOUR website! . the sum of a number and 9 is multiplied by -2 and the answer is -8 find the number ~~~~~~~~~~~~~~~~~~~~~~ The simplest way to solve it is to make the operations in the reverse order: 1. = 4. 2. Hence, the number plus 9 is 4. 3. Hence, the number is 4 - 9 = -5.

Question 1063666 : The sum of two numbers is 59 The larger number is 8 more than the smaller number. What are the numbers? L - S = 8 ------------- Add Question 1063556 : On Day 1, you have $3 in your wallet. On the morning of day 2, you count your money and are astonished to find that there is now $5 in your wallet. On the morning of day 3, you count your money and now there is $7 in your wallet. On day 4, you are delighted to find that you now have $9 in your wallet. This continues every day and at the beginning of each day, you count your money and find that an additional $2 has magically appeared in your wallet. Assume that you do not spend any of the money in your wallet (and that your wallet is really large and can hold a lot of $1 bills). How much money is in your wallet on the morning of day 10? on the morning of day 100? Challenge: What if "n" represents any day. Can you find the amount of money in your wallet on Day "n"?

You can put this solution on YOUR website! On Day 1, you have $3 in your wallet. On the morning of day 2, you count your money and are astonished to find that there is now $5 in your wallet. On the morning of day 3, you count your money and now there is $7 in your wallet. On day 4, you are delighted to find that you now have $9 in your wallet. This continues every day and at the beginning of each day, you count your money and find that an additional $2 has magically appeared in your wallet. Assume that you do not spend any of the money in your wallet (and that your wallet is really large and can hold a lot of $1 bills). How much money is in your wallet on the morning of day 10? on the morning of day 100? Day 1:: 3 Day 2:: 3+2 = 5 Day 3:: Day 4:: 3+2+2+2 = 9 Day 10:: 3 + 9*2 = 3+18 = 21 ------ Challenge: What if "n" represents any day.

Can you find the amount of money in your wallet on Day "n"? Day n:: 3 + (n-1)2 = 3 + 2n-2 = 2n+1 --------------- You can put this solution on YOUR website! 8 - 4x Question 1063389 : The sum of the digits of two digit number is 15. One third of the number diminished by 12 is 17. What is the number? Show your solution. You can put this solution on YOUR website! . The sum of the digits of two digit number is 15. One third of the number diminished by 12 is 17. What is the number? Show your solution. ~~~~~~~~~~~~~~~~~~~~~~~~ The second condition says ("One third of the number diminished by 12 is 17.") = 17 ---> x - 36 = 51 ---> x = 51 + 36 ---> x = 87. The first condition is unnecessary and does not used in the solution. Happily, it is consistent with the found solution. Answer. The number is 87.

Question 1063388 : The sum of the digits of a two digit number is 15. One third of the number diminished by 12 is 17. What is the number. Show your solution Found 2 solutions by josmiceli, josgarithmetic: Digit sum equation t+u=15 Number is 87 Question 1063346 : 3+5+6=151872,5+5+6=253094,5+6+7=303585,5+5+3=251573 than What is 9+4+7= Found 2 solutions by Edwin McCravy, ikleyn: You can put this solution on YOUR website!

The first part of the pattern is easy, if you separate the digits on the right: 3+5+6=15 18 72 5+5+6=25 30 94 5+6+7=30 35 85 5+5+3=25 15 73 The number formed by the 1st and 2nd digits on the right is the product of the first two numbers on the left. The number formed by the 3rd and 4th digits on the right is the product of the first and third numbers on the left. So we know eveything but the last two digits: 9+4+7=3663?? We only have to figure out a pattern to go from the 4 numbers 3,5,6,15,18 to 72, 5,5,6,25,30 to 94, 5,6,7,30,35 to 85, 5,5,3,25,15 to 73.

The sum of the last two numbers are 15+18 = 33, not close to 72, but noticeably close to its reverse 27 25+30 = 55, not close to 94, but noticeably close to its reverse 49 30+35 = 65, not close to 85, but noticeably close to its reverse 58 25+15 = 40, not close to 73, but noticeably close to its reverse 37 Let's see how close they are to those reverses? 33 is 6 more than 27 55 is 6 more than 49 65 is 7 more than 58 40 is 3 more than 37 Now we notice that 6,6,7,3 are the third numbers. So the pattern is 1st 4 digits on the right = the reverse of: the sum of (the product of the 1st and 2nd on left) and (the product of the 3rd and 4th on left), decreased by the 3rd digit.

So for 9+4+7=3663?? We add 36+63, get 99, then we subtract the 3rd digit on the left, 7, and get 92. Then we turn it backward as 27. Edwin The sum of digits of a two-digit number is 9.

On interchanging the order of the digits,the number becomes 27 more then the original number.find the number Let a be the 10's digit and b be the 1's digit The number is 10a+b (1) a+b = 9 (given) (2) 10b+a = (10a+b)+27 (interchanging digits results in a number that is 27 greater than original) (1) ==> a=9-b Substitute for '9-b' for 'a' in (2), that will give us an equation with just 'b' and we can solve for 'b' 10b + (9-b) = (10(9-b) + b) + 27 9b + 9 = 90 - 10b + b + 27 9b + 9 = 117 - 9b IP58d: Food and Agriculture: Consumer Trends and Opportunities, Fruits Food and Agriculture: Consumer Trends and Opportunities Fruits Betty S. King, Extension Specialist in Rural Economic Development Janet L. Tietyen, Extension Specialist in Food and Nutrition Steven S.

Vickner, Assistant Professor in Agricultural Economics This publication describes the trends in consumption, nutrition, health, lifestyle, and marketing for the fruit sector of the agricultural economy. This publication is part of a series that seeks to integrate the consumer aspects of food and agriculture in an effort to help Kentucky fruit farmers. Each publication is organized around the USDA's Food Guide Pyramid. The series is designed to bridge gaps in understanding about the economics of food consumption, health and lifestyle trends, and food production and to provide a resource for food marketing efforts. The following information should be helpful for farmers who want to better understand consumers and their consumption patterns. Consumers may gain a better understanding of the nutritional implications of their diet. Per Capita Fruit Consumption Trends Which fruit products do consumers purchase? Figure 1 shows the trends in domestic per capita consumption (in pounds) for both fresh and processed fruit from 1970 to 1995. These data are based on the disappearance1of the fruit supply, rather than actual fruit consumption. Generally, disappearance data overestimate consumption. However, by keeping track of disappearance trends over time, researchers can determine relative changes in fruit products consumed.

Of the 280.9 pounds of fruit consumed per capita domestically in 1995, 55 percent was processed and 45 percent was fresh. This mix has remained fairly constant since 1970 as the consumption of both processed and fresh fruit has increased. Processed fruit consumption, while reaching its peak of 163.7 pounds in 1977, increased on average 21 percent from 127.8 to 154.8 pounds between 1970 and 1995. Fresh fruit consumption during the same period increased 25 percent, ending 1995 at 126.1 pounds. The mix of fresh fruit consumption has changed markedly through time. Although it was roughly one-third of the fresh fruit consumed in 1970, citrus fruit consumption had fallen to only 19 percent of the mix in 1995. Figure 2 shows that citrus fruit consumption fell 16 percent from 28.9 to 24.4 pounds per capita during the analysis period, reaching its minimum at 19.1 pounds in 1991.

Since 1970, fresh banana and melon consumption increased 57 percent and 31 percent, respectively. While the 25-year consumption trends have mirrored each other, in 1995 Americans ate 28.2 pounds of fresh melons, slightly more (0.8 pounds) than bananas. Fresh apple consumption fluctuated during the last 25 years, ending 1995 slightly above its 1970 level at 18.9 pounds. Total consumption of processed non-citrus fruit has remained fairly constant at roughly two-fifths of all processed fruit consumed domestically, but its mix has changed. Figure 3 illustrates a rapid increase in the per capita consumption of processed apples, up 88 percent from 1970 to 27.4 pounds in 1995. The consumption of fruits identified in this publication reached its peak of 30.1 pounds in 1994. Processed pineapple consumption on average remained around 11.0 pounds since 1970. Changes in relative prices, income, preferences, and the marketing mix have largely shaped these trends.

Other Forces Driving Fruit Consumption More than half of all American consumers eat some type of fruit or drink fruit juice nearly every day. Children age 5 and younger and older adults age 70 and older eat the most fruit. Consumer taste preferences and innovations in convenience packaging are influencing the consumption and marketing mix of fruits. Bananas, melons, citrus fruits, and apples continue to be popular choices. Non-citrus juices and juice drinks, which are apple or grape juice-based, contributed in large part to the increased consumption of fruit, up by 20 percent during the last 25 years. Although a greater number of Americans are eating away from home and consuming more processed foods, there are opposing trends as well. For example, the number of small specialty food stores and farmers markets is increasing along with the size of supermarkets. Market research studies identified consumer attitude changes toward environmental issues regarding food safety and production. Consumers are seeking more information from food labeling to guide decisions about their food choices. While many fruits, such as bananas, are imported to the United States, consumers have many opportunities to frequent local markets for seasonal fruits. On-farm roadside markets, pick-your-own operations, and farmers' markets offer fresh, seasonal fruits and berries.

Supermarket stores and specialty health, gourmet, and ethnic food stores also offer both common and exotic fruits. Consumers can benefit from the variety of fresh fruits and berries at both retail and direct market outlets. Examples of Fruit Marketing Kentucky farmers produced more than 13.6 million pounds of commercial apples and 600,000 pounds of commercial peaches in 1996, totaling more than $4.5 million in value. Wholesale and direct markets such as pick-your-own operations or on-farm roadside markets are the most common market forms of fruit sales in Kentucky. In addition to fresh fruit sales, a number of value-added fruit products such as jams and jellies are listed in the Kentucky Food Products directory. Pick-your-own operations have been declining in recent years due largely to consumers' time constraints. A certified roadside market program was developed by the Kentucky Farm Bureau and is promoted through the Kentucky agriculture and tourism promotion programs. Many roadside and on-farm markets are diversifying their activities to include agri-entertainment activities such as corn mazes, hay rides, and mini-festivals for children and adults to attract potential buyers to their operations. Thirty-seven certified roadside markets are listed in the Kentucky Farm Bureau promotional brochure.

One example of how farmers are marketing fruits through roadside markets is Garrett's Orchard and Country Market, located in Versailles, Kentucky. It features apples, peaches, blueberries, raspberries, strawberries, and other fruits and other farm produce. In its country market, the company sells Kentucky value-added products such as gift baskets of molasses, bakery products, and Kentucky ham. It also provides school tours, educational seminars, afternoon teas, a petting zoo, and picnic areas. Several mini-festivals are conducted each year, including a corn, apple, and pumpkin festival. Fruit Nutrition and Health U.S. food consumers are currently eating only half the amount of fruit recommended in the USDA Food Guide Pyramid. The minimum number of servings recommended is two per day, but in 1995, nearly half of Americans consumed less than one fruit serving per day. The trend toward more meals eaten away from home is consistent with low fruit consumption, as many restaurants may not offer fruit. Perhaps one of the most convenient ways for food consumers to increase the amount of fruit in their diets is to simply carry fresh fruit with them to work or school. Some consumers are evidently following this advice.

From 1982 to 1997, fresh fruit consumption increased 24 percent in the United States. Fruits contribute important nutrients to the diet, including vitamins A and C, folate, potassium, and dietary fiber. The potential health benefits of fruit and vegetable consumption are featured in the "5-A-Day for Better Health" campaign, which recommends that consumers eat five servings from these foods to lower disease risk and promote optimal health. In 1970, vegetables were the leading source of vitamin C in the U.S. diet, contributing 50 percent of the supply, while fruits contributed 39 percent. By 1994, fruits supplied a greater proportion of vitamin C, contributing 44 percent of the national supply. Fruits contribute 12 percent of the folate, a nutrient now recognized for its role in reducing the risk of heart disease and cancer, and they contribute 12 percent of potassium, a mineral with beneficial effects on blood pressure. Fruits contributed 3.4 percent of calories to the U.S. food supply in 1994, up from 2.9 percent in 1970.

Current USDA research at the Human Nutrition Research Center on Aging at Tufts University promises to reveal more good nutrition news about fruit in the U.S. diet (USDA-ARS, 1999). Fruits such as prunes, raisins, blueberries, blackberries, strawberries, and raspberries are proving to have potent antioxidant capacity. Foods such as fruits and vegetables with antioxidants may help decrease risk of heart disease, stroke, and cancer. People who eat fruits tend to have healthier diets. In the period from 1994 to 1996, women who ate no fruit on one day consumed only 82 percent of the recommended amount of vitamin C. Women who included fruit in their diet consumed 205 percent of the vitamin C recommended. Researchers have also observed that women who eat fruit tend to have a lower fat intake than those who do not eat fruit (31 percent versus 34 percent of calories from fat). Fresh fruits such as apples, bananas, oranges, and peaches are appealing to today's food consumer because they can be eaten just about anywhere or anytime. There are more than 60 fruit products included in the U.S.

food supply data, suggesting that many consumers would benefit from more variety in daily fruit choices. Nearly half of total fruit servings in the U.S. diet came from five foods in 1994: orange juice (18 percent) apple juice (5.8 percent). What You Can Do The food choices we make within our food and agricultural system impact our local, state, and national economy, the environment, and the well-being of communities, as well as our own personal health. Here are some practical things you can do: As a Consumer Learn more about your community food system and the origins of your food purchases. Determine if your neighborhood restaurant and grocery stores use and sell Kentucky produce. Visit a roadside on-farm market to learn more about fresh produce. As a Community Leader Work with your local Cooperative Extension Service office or chamber of commerce to promote local agricultural educational and economic development activities. As a Farmer Learn about your customers' needs and wants. Educate consumers about farming and the farm products you market. Join community organizations that foster interest and support in farming. Learn more about the costs and benefits of value-added agricultural opportunities.

1 This term, as defined by the USDA-ERS, means beginning food stocks, production, and imports minus exports, shipments to the U.S. territories, and ending stocks. So it is a reasonable proxy for consumption, given that data for consumption is not collected overall. References 1997 Kentucky Certified Roadside Farm Markets Directory. Kentucky Farm Bureau, Commodity Relations Department, Louisville, Ky. Borrud, L., C. W. Enns, and S. Mickle. "What We Eat in America: USDA Surveys Food Consumption Changes. FoodReview. USDA Economic Research Service, Washington, D.C., September-December, 1996. Putnam, J., and J. Allshouse. Food Consumption, Prices, and Expenditures 1970-95. USDA Economic Research Service, Washington, D.C., Statistical Bulletin No. 939, August 1997. USDA, Agricultural Research Service. Pyramid Servings Data: Results from USDA's 1995 and 1996 Continuing Survey of Food Intake by Individuals. Washington, D.C., 1997. USDA, Agricultural Research Service.

Top Antioxidant Foods. Washington, D.C., February 1999. USDA, Economic Research Service. America's Eating Habits: Changes and Consequences. E. Frazao, ed. Washington, D.C., Agricultural Information Bulletin No. 750, 1997. USDA, Economic Research Service. Fruit and Tree Nuts Situation and Outlook Report. Washington, D.C., March 1999. No special endorsement of the companies named in this publication is implied. Additional Extension publications are available in the Food and Agriculture: Consumer Trends and Opportunities series. Ask your county Extension office for these publications multiplication_and_division return to top Assumed Knowledge Much of the building of understanding of early mathematics occurs concurrently, so a child can be developing the basic ideas related to multiplication and division whilst also investigating the place-value system. However, there are some useful foundations necessary for multiplication and division of whole numbers: Some experience with forwards and backwards skip-counting. Some experience doubling and halving small numbers. return to top Motivation One way of thinking of multiplication is as repeated addition. Multiplicative situations arise when finding a total of a number of collections or measurements of equal size. Arrays are a good way to illustrate this.

Some division problems arise when we try to break up a quantity into groups of equal size and when we try to undo multiplications. Multiplication answers questions such as: 1 Judy brought 3 boxes of chocolates. Each box contained 6 chocolates. How many chocolates did Judy have? 2 Henry has 3 rolls of wire. Each roll is 4m long. What is the total length of wire that Henry has? Division answers questions such as: 1 How many apples will each friend get if four friends share 12 apples equally between them? 2 If twenty pens are shared between seven children how many does each child receive, and how many are left over? Addition is a useful strategy for calculating ‘how many’ when two or more collections of objects are combined. When there are many collections of the same size, addition is not the most efficient means of calculating the total number of objects. For example, it is much quicker to calculate 6 × 27 by multiplication than by repeated addition. Fluency with multiplication reduces the cognitive load in learning later topics such as division. The natural geometric model of multiplication as rectangular area leads to applications in measurement. As such, multiplication provides an early link between arithmetic and geometry.

Fluency with division is essential in many later topics and division is central to the calculations of ratios, proportions, percentages and slopes. Division with remainder is a fundamental idea in electronic security and cryptography. return to top Content Multiplication and division are related arithmetic operations and arise out of everyday experiences. For example, if every member of a family of 7 people eats 5 biscuits, we can calculate 7 × 5 to work out how many biscuits are eaten altogether or we can count by ‘fives’, counting one group of five for each person. In many situations children will use their hands for multiples of five. For whole numbers, multiplication is equivalent to repeated addition and is often introduced using repeated addition activities. It is important, though that children see multiplication as much more than repeated addition. If we had 35 biscuits and wanted to share them equally amongst the family of 7, we would use sharing to distribute the biscuits equally into 7 groups. We can write down statements showing these situations: 7 × 5 = 35 and 5 × 7 = 35 Also, Introducing vocabulary and symbols There is a great deal of vocabulary related to the concepts of multiplication and division.

For example, multiplication − multiply, times, product, lots of, groups of, repeated addition division − sharing, divided by, repeated subtraction Some of these words are used imprecisely outside of mathematics. For example, we might say that a child is the product of her environment or we insist that children ‘share’ their toys even though we do not always expect them to share equally with everyone. It is important that children are exposed to a variety of different terms that apply in multiplication and division situations and that the terms are used accurately. Often it is desirable to emphasise one term more than others when introducing concepts, however a flexibility with terminology is to be aimed for. Looking at where words come from gives us some indication of what they mean. The word ‘multiply’ was used in the mathematical sense from the late fourteenth century and comes from the Latin multi meaning ‘many’ and plicare meaning ‘folds’ giving multiplicare - ‘having many folds’, which means ‘many times greater in number’. The term ‘manyfold’ in English is antiquated but we still use particular instances such as ‘twofold’ or threefold’. The word ‘divide’ was used in mathematics from the early 15th century.

It comes from the Latin, dividere meaning ‘to force apart, cleave or distribute’. Interestingly, the word widow has the same etymological root, which can be understood in the sense that a widow is a woman forced apart from her husband. Use of the word ‘product’ The product of two numbers is the result when they are multiplied. So the product of 3 and 4 is the multiplication 3 × 4 and is equal to 12. It is important that we use the vocabulary related to multiplication and division correctly. Many years ago we were told to ‘do our sums’ and this could apply to any calculation using any of the operations. This is an inaccurate use of the word ‘sum’. Finding the ‘sum’ of two or more numbers means to add them together. Teachers should take care not to use the word ‘sum’ for anything but addition. The symbols × and ÷ The × symbol for multiplication has been in use since 1631. It was chosen for religious reasons to represent the cross. We read the statement 3 × 4 as ‘3 multiplied by 4’. In some countries a middle dot is used so 3 × 4 is written as 3.4.

In algebra it is common to not use a symbol for multiplication at all. So, a × b is written as ab. The division symbol ÷ is known as the obelus. It was first used to signify division in 1659. We read the statement 12 ÷ 3 as ‘12 divided by 3’. Another way to write division in school arithmetic is to use the notation , meaning ‘12 divided by 3’, but sometimes read as ‘3 goes into 12’. Mathematicians almost never use the ÷ symbol for division. Instead they use fraction notation. The writing of a fraction is really another way to write division. So 12 ÷ 4 is equivalent to writing , where the numerator, 12, is the dividend and the denominator, 4, is the divisor. The line is called called a vinculum, which is a Latin word meaning ‘bond or link’. Once students are becoming fluent with the concepts of multiplication and division then the symbolic notation, × for multiplication and ÷ for division, can be introduced.

Initially, the ideas will be explored through a conversation, then written in words, followed by a combination of words and numerals and finally using numerals and symbols. At each step, when the child is ready, the use of symbols can reflect the child’s ability to deal with abstract concepts. Modelling multiplication by arrays Rectangular arrays can be used to model multiplication. For example, 3 × 5 is illustrated by We call 15 the product of 3 and 5, and we call 3 and 5 factors of 15. By looking at the rows of the array we see that 3 × 5 = 5 + 5 + 5 By looking at the columns of the array we also see that 5 × 3 = 3 + 3 + 3 + 3 + 3 This illustrates 3 × 5 = 5 × 3. We say that multiplication is commutative. Arrays are useful because they can be used with very small as well as very large numbers, and also with fractions and decimals. CLASSROOM ACTIVITY Children can model multiplication using counters, blocks, shells or any materials that are available and arranging them in arrays.

1 Describe the multiplication using words, words and numbers and finally words and symbols. Modelling multiplication by skip-counting and on the number line Skip-counting, such as reciting 3, 6, 9, 15,..., is one of the earliest introductions to repeated addition and hence to multiplication. This can be illustrated on a number line as shown for 3 × 5 = 15 below. 3 × 5 = 15 On the number line, the fact that 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 is not so obvious; the previous image shows 5 + 5 + 5, whereas 3 + 3 + 3 + 3 + 3 looks quite different. Skip-counting is important because it helps children learn their multiplication tables. Modelling multiplication by area Replacing objects in an array by unit squares provides a natural transition to the area model of multiplication. This is illustrated below for 3 × 5. At this stage, we are simply using unit squares instead of counters or stars. We can also use the area model of multiplication later for multiplication of fractions.

return to top Properties of Multiplication One of the advantages of the array and area approach is that properties of multiplication are more apparent. Commutativity As discussed above, turning the 3 × 5 array on its side illustrates that 3 × 5 = 5 × 3 because the total number of objects in the array does not change. 3 × 5 = 5 × 3 We saw this before by looking at the rows and columns separately, but we can also do this by turning the rectangle on its side. The area of the rectangle does not change. 3 × 5 = 5 × 3 Another important property of multiplication is associativity, which says that a × (b × c) = (a × b) × c for all numbers. We can demonstrate this with the numbers 2, 3 and 4: 2 × (3 × 4) = (2 × 3) × 4 Associativity of multiplication ensures that the expression a × b × c is unambiguous. Any-order property We usually don’t teach young children associativity of multiplication explicitly when introducing multiplication. Instead, we teach the any-order property of multiplication, which is a consequence of the commutative and associative properties.

Any-order property of multiplication A list of numbers can be multiplied together in any order to give the product of the numbers. The any-order property of multiplication is analogous to the any- order property of addition. Both associativity and commutativity are nontrivial observations; note that subtraction and division are neither commutative nor associative. Once we are familiar with the arithmetic operations we tend to take both associativity and commutativity of multiplication for granted, just as we do for addition. Every so often, it is worth reflecting that commutativity and associativity combine to give the important and powerful any-order properties for addition and multiplication. Multiplying three whole numbers corresponds geometrically to calculating the number of unit cubes in (or volume of) a rectangular prism. The any-order property of multiplication means that we can calculate this volume by multiplying the lengths of the sides in any order. The order of the calculation corresponds to slicing the volume up in different ways. 5 × 2 = 2 × 5 (5 × 3) × 2 = (3 × 5) × 2 We can apply this to the multiplication of three or more numbers, it doesn’t matter in which order we do this.

Distributivity of Multiplication over Addition The equation 3 × (2 + 4) = (3 × 2) + (3 × 4) is an example of the distributivity of multiplication over addition. With arrays, this corresponds to the following diagram. With areas it corresponds to the diagram below. Multiplication is also distributive over subtraction. For example 7 × (10 − 2) = 7 × 10 − 7 × 2. We use the distributive property to enable us to reduce multiplication problems to a combination of familiar multiples.

For example, 7 × 101 = 7 × (100 + 1) = 700 + 7 = 707, 7 × 99 = 7 × (100 − 1) = 700 − 7 = 693, 7 × 102 = 7 × (100 + 2) = 700 + 14 = 714, and 7 × 98 = 7 × (100 − 2) = 700 − 14 = 686. EXERCISE 1 The effect of multiplying by one When any number is multiplied by 1, the number is unchanged. For example, 5 × 1 = 5 = 1 × 5 We call 1 the multiplicative identity. It is important to have this conversation with young children in very simple terms, using lots of examples in the early stages of developing understanding about multiplication. Zero is the identity element for addition. When nothing is added to a set there is no effect on the number of objects in that set.

For example, 5 + 0 = 5 = 0 + 5. This is true for all addition. Hence, we call zero the identity element for addition of whole numbers. The effect of multiplying by zero When any number is multiplied by zero the result is zero. Situations showing the effect of multiplying by zero can be acted out with children using concrete objects. For example, If I have 5 baskets with three apples in each I have 5 × 3 = 15 apples in total. However, if I have 5 baskets with 0 apples in each, the result is 5 × 0 = 0 apples in total. return to top Learning the multiplication table. Fluency with multiplication tables is essential for further mathematics and in everyday life. For a while it was considered unnecessary to learn multiplications tables by memory, but it is a great help to be fluent with tables in many areas of mathematics. If students can add a single-digit number to a two-digit number, they can at least reconstruct their tables even if they have not yet developed fluency. It is therefore essential to ensure that students can add fluently before they begin to learn their ‘tables’.

We strongly recommend that students learn their multiplication facts up to 12 × 12. This is primarily because the 12 times table is essential for time calculations — there are 12 months in a year, 24 hours in a day, and 60 minutes in an hour. Familiarity with dozens is useful in everyday life because packaging in 3 × 4 arrays is so much more convenient than in 2 × 5 arrays. In addition, the 12 × 12 table has many patterns that can be constructively exploited in pre-algebra exercises. A straightforward approach to learning the tables is to recite each row, either by heart or by skip-counting. However, students also need to be able to recall individual facts without resorting to the entire table. Looking at the 12 × 12 multiplication table gives the impression that there are 144 facts to be learnt. However, there are several techniques that can be used to reduce the number of facts to be learnt. The commutativity of multiplication (8 × 3 = 3 × 8) immediately reduces this number to 78.

The 1 and 10 times tables are straightforward and their mastery reduces the number of facts to be learnt to 55. The 2 and 5 times tables are the easiest to learn and their mastery further reduces the number of facts to be learnt to 36. The 9 and 11 times tables are the next easiest to skip-count because 9 and 11 differ from 10 by 1. This reduces the number of facts to 21. Children may notice the decreasing ones digit and increasing tens digit in the nine times table. They may also be intrigued by the fact that the sum of the digits of a multiple of 9 is always 9. The squares are useful and can be learnt just as one might learn a times table. This reduces the number of terms to be learnt to 15. Whatever techniques are used, the aim should be fluency. return to top MODELLING DIVISION Division always involves splitting something into a number of equal parts, but there are many contrasting situations that can all be described by ‘division’.

Before introducing the standard algorithm for division, it is worthwhile discussing some of these situations under the headings: Division without remainder, Division without remainder Here is a simple model of the division 24 ÷ 8. Question: If I pack 24 apples into boxes, each with 8 apples, how many boxes will there be? We can visualise the packing process by laying out the 24 apples successively in rows of 8, as in the diagrams below. The 3 rows in the last array use up all 24 apples, so there will be 3 full boxes, with no apples left over. The result is written in mathematical symbols as The number 24 is called the dividend (‘that which is to be divided’). The number 8 is called the divisor (‘that which divides’). The number 3 is called the quotient, (from the Latin quotiens meaning ‘how many times’). Modelling division by skip-counting and on the number line Division without remainder can be visualized as skip-counting. 0, 8, 16, 24… On the number line we count in 8s until we reach 24.

EXERCISE 2 Evaluate 1000 ÷ 100 by counting in 100s. Using arrays to show division without remainder is the inverse of multiplication The rectangular array that we produced when we modeled 24 ÷ 8 is exactly the same array that we would draw for the multiplication 3 × 8 = 24. In our example: The statement 24 = 8 × 3 means ‘three boxes, each with 8 apples, is 24 apples’, and The statement 24 ÷ 8 = 3 means ‘24 apples make up 3 boxes, each with 8 apples’. Division without remainder is the inverse process of multiplication. The multiplication statement 24 = 8 × 3 can in turn be reversed to give a second division statement 24 ÷ 3 = 8 which answers the question, ‘What is 24 divided by 3? This corresponds to rotating the array by 90°, and regarding it as made up of 8 rows of 3.

It answers the question, ‘If I pack 24 apples into boxes each holding 3 apples, how many boxes will be required?’ So the division statement 24 ÷ 8 = 3 now has four equivalent forms: 24 ÷ 8 = 3 and 24 = 8 × 3 and 24 = 3 × 8 and 24 ÷ 3 = 8. EXERCISE 3 81 ÷ 9 = 9. What happened in this example, and why? Two models of division without remainder This section is included for teachers because children’s questions often concern pairs of situations similar to those described here. If we have 24 balloons to share equally, there are two ways we can share them. The first way is by asking ‘How many groups?’ For example, if we have 24 balloons and we give 8 balloons each to a number of children, how many children get 8 balloons? If we split 24 balloons into groups of 8, then 3 children get 8 balloons each. We say ‘24 divided by 8 is 3’. This is written as 24 ÷ 8 = 3.

We can see this from the array: 3 lots of 8 make 24 24 ÷ 8 = 3 The second way is by asking ‘How many in each group?’ For example, if we share 24 balloons among 8 children, how many balloons does each child receive? We want to make 8 equal groups. We do this by handing out one balloon to each child. This uses 8 balloons. Then we do the same again. We can do this 3 times, so each child gets 3 balloons. Again, we can see this from the multiplication array: So dividing 24 by 8 is the same as asking ‘Which number do I multiply 8 by to get 24?’ For each division problem, there is usually an associated problem modelling the same division statement. The ‘balloons’ example above shows how two problems can have the same division statement. One problem with balloons is the associate of the other. EXERCISE 4 Write down in symbols the division statement, with its answer, for each problem below. Then write down in words the associated problem: a Division with remainder We will now use apples to model 29 ÷ 8.

Question: If I pack 29 apples into boxes, each with 8 apples, how many boxes will there be? As before, we can visualise the packing process by laying out the 29 apples successively in rows of 8: We can lay out 3 full rows, but the last row only has five apples, so there will be 3 full boxes and 5 apples left over. The result is written as dividend divisor quotient remainder The number 5 is called the remainder because there are 5 apples left over. The remainder is always a whole number less than the divisor. As with division without remainder, skip-counting is the basis of this process: 0, 8, 16, 24, 32,… We locate 29 between successive multiples 24 = 8 × 3 and 32 = 8 × 4 of the divisor 8. Then we subtract to find the remainder 29 − 24 = 5.

We could also have answered the question above by saying, ‘There will be four boxes, but the last box will be 3 apples short.’ This corresponds to counting backwards from 32 rather than forwards from 24, and the corresponding mathematical statement would be 29 ÷ 8 = 4 remainder (−3). It is not normal practice at school, however, to use negative remainders. Even when the question demands the interpretation corresponding to it, we will always maintain the usual school convention that the remainder is a whole number less than the divisor. Division without remainder can be regarded as division with remainder 0. During the location process, we actually land exactly on a multiple instead of landing between two of them. For example, 24 ÷ 8 = 3 remainder 0, or more simply, 24÷ 8 = 3, and we say that 24 is divisible by 8 and that 8 is a divisor of 24. The corresponding multiplication and addition statement The 29 apples in our example were packed into 3 full boxes of 8 apples, with 5 left over.

We can write this as a division, but we can also write it using a product and a sum, 29 ÷ 8 = 3 remainder 5 or 29 = 8 × 3 + 5 So for division with remainder there is a corresponding statement with a multiplication followed by an addition, which is more complicated than division without remainder. Two models of division with remainder As before, problems involving division with remainder usually have an associated problem modelling the same division statement. Continuing with our example of 29 ÷ 8 = 3 remainder 5: Question: How many bags of 8 apples can I make from 29 apples and how many are left over? Question: I have 29 apples and 8 boxes. How many apples should I put in each box so that there is an equal number of apples in each box and how many are leftover? The following two associated questions model 63 ÷ 10 = 6 remainder 3. Question: If I have 63 dollar coins, and ten people to give them to, how many coins does each person get if they are to each have the same number of coins? How many are left over?

Question: If I have 63 dollar coins, how many $10 books can I buy and how many dollars do I have left over? EXERCISE 5 Order and brackets cannot be ignored When multiplying two numbers, the order is unimportant. For example, 3 × 8 = 8 × 3 = 24. When dividing numbers, however, the order is crucial. For example, 20 ÷ 4= 5, but 4 ÷ 20 = To visualise this calculation, 20 people living in 4 homes means each home has on average 5 people, whereas 4 people living in 20 homes means each home has on average of a person. Similarly when multiplying numbers, the use of brackets is unimportant. For example, (3 × 4) × 5 = 12 × 5 = 60 and 3 × (4 × 5) = 3 × 20 = 60. When dividing numbers, however, the use of brackets is crucial.

For example, (24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3; but 24 ÷ (4 ÷ 2)= 24 ÷ 2= 12 Division by zero Earlier we used empty baskets of apples to illustrate that 5 × 0 = 0. The same model can be used to illustrate why division by zero is undefined. If we have 10 apples to be shared equally amongst 5 baskets each basket will have 10 ÷ 5 = 2 apples in each. If the 10 apples are shared equally between 10 baskets, each basket has 10 ÷ 10 = 1 apples in each. If 10 apples are shared between 20 baskets, each basket will have an apple in each. What happens if we try to share 10 apples between 0 baskets? This cannot be done. If 10 ≠ a × 0. This action is meaningless, so we say that 10 ÷ 0 is undefined.

We must always be careful to relate this to children accurately so that they understand that: 10 ÷ 0 is NOT equal to 1 and 10 ÷ 0 is NOT equal to 0 but 10 ÷ 0 is not defined. Dividing by 4, 8, 16, . . . Because 4 = 2 × 2 and 8 = 2 × 2 × 2, we can divide by 4 and 8, and by all powers of 2, by successive halving. To divide by 4, halve and halve again. For example, to divide 628 by 4, 628 ÷ 4 = (628 ÷ 2) ÷ 2 = 314 ÷ 2 = 157 To divide by 8, halve, halve, and halve again.

For example, to divide 976 by 8, 976 ÷ 8 = (976 ÷ 2) ÷ 2 ÷ 2 = 488 ÷ 2 ÷ 2 = 244 ÷ 2 = 122 EXERCISE 6 return to top Multiplication Algorithm An algorithm works most efficiently if it uses a small number of strategies that apply in all situations. So algorithms do not resort to techniques, such as the use of near-doubles, that are efficient for a few cases but useless in the majority of cases. The standard algorithm will not help you to multiply two single-digit numbers. It is essential that students are fluent with the multiplication of two single-digit numbers and with adding numbers to 20 before embarking on any formal algorithm. The distributive property is at the heart of our multiplication algorithm because it enables us to calculate products one column at a time and then add the results together. It should be reinforced arithmetically, geometrically and algorithmically.

For example, arithmetically we have 6 × 14 = 6 × 10 + 6 × 4, geometrically we see the same phenomenon, and algorithmically we implement this in the following calculation. 1 Once this basic property is understood, we can proceed to the contracted algorithm. Introducing the algorithm using materials Initially when children are doing multiplication they will act out situations using blocks. Eventually the numbers they want to multiply will become too large for this to be an efficient means of solving multiplicative problems. However base-10 materials or bundles of icy-pole sticks can be used to introduce the more efficient method - the algorithm. If we want to multiply 6 by 14 we make 6 groups of 14 (or 14 groups of 6): Collect the ‘tens’ together and collect the ‘ones’ together. This gives 6 ‘tens’ and 24 ‘ones’. Then make as many tens from the loose ones. There should never be more than nine single ones when representing any number with Base-10 blocks. This gives 6 ‘tens’ + 2 ‘tens’ + 4 ‘ones’.

We add the tens to get 14 × 6 = 10 × 6 + 4 × 6 = 60 + 20 + 4 = 84 Eventually we should start recording what is being done with the blocks using the multiplication algorithm vertical format. Eventually the support of using the blocks can be dropped and students can complete the algorithm without concrete materials. Multiplying by a single digit First we contract the calculation by keeping track of carry digits and incorporating the addition as we go. The previous calculation shortens as either or depending on where the carry digits are recorded. Care should be taken even at this early stage because of the mixture of multiplication and addition. Note also that the exact location and size of the carry digit is not essential to the process and varies across cultures. Multiplying by a single-digit multiple of a power of ten The next observation is that multiplying by a single-digit multiple of ten is no harder than multiplying by a single digit provided we keep track of place value.

So, to find the number of seconds in 14 minutes we calculate 14 × 60 = 14 × 6 × 10 = 840 and implement it algorithmically as 2 corresponds to the area decomposition It is not efficient to do this extended long multiplication in order to calculate products in general, but it can be used to highlight the multiple use of distributivity in the process. The area model illustration used in this case reappears later as a geometric interpretation of calculations in algebra. return to top The standard division algorithm There is only one standard division algorithm, despite its different appearances. The algorithm can be set out as a ‘long division’ calculation to show all the steps, or as a ‘short division’ algorithm where only the carries are shown, or with no written working at all.

Setting the calculation out as a long division We could set the work out as follows: 5 × 400 = 2000, then subtract 2000 from 2193 5 × 30 = 150, then subtract 150 from 193 5 × 8 = 40, then subtract 40 from 43 The standard ‘long division’ setting-out, however, allows place value to work for us even more efficiently, by working only with the digits that are required for each particular division. At each step another digit is required − this is usually called ‘bringing down the next digit’. Divide 21 by 5. 5 × 4 = 20, then subtract 20 from 21. Bring down the 9, and divide 19 by 5. 5 × 3 = 15, then subtract 15 from 19. Bring down the 3, and divide 43 by 5. 5 × 8 = 40, then subtract 40 from 43.

Hence 2193 ÷ 5 = 138 remainder 3. (Never forget to gather the calculation up into a conclusion.) The placing of the digits in the top line is crucial. The first step is ‘5 into 21 goes 4’, and the digit 4 is placed above the digit 1 in 21. Setting the calculation out as a short division Once the steps have been mastered, many people are comfortable doing each multiplication/subtraction step mentally and writing down only the carry. The calculation then looks like this: We say, We can see that 350 minutes = 300 minutes + 50 minutes = 5 hours and 50 minutes. With a calculator using the division key: Enter 350 ÷ 60, and the answer is 5.833333… hours. Then subtract 5 to get 0.833333…, and multiply by 60 to convert to 50 minutes, giving the answer 5 hours and 50 minutes.

Calculator assistance may be extremely useful with larger numbers, but experience with long division is essential to interpret the calculator display This phenomenon is common to many similar situations in mathematics. return to top Links Forward The first application of multiplication that students are likely to meet is division. When calculating a division, we are constantly calculating multiples of the divisor, and lack of fluency with multiplication is a significant handicap in this process. The material in this module lays the foundation for multiplication, and then division, of fractions and decimals. Other applications of multiplication include percentages and consumer arithmetic. For example, we calculate the price of an item inclusive of GST by calculating 1.1 times its pre-GST cost. A familiarity with multiplication and the expression of numbers as products of factors paves the way for one of the major theorems in mathematics. The Fundamental Theorem of Arithmetic states that every whole number bigger than 1 can be written as a product of prime numbers and such an expression is unique up to the order in which the factors are written. For example, 24 = 23 × 3 and 20 = 22 × 5. The Fundamental Theorem of Arithmetic has far-reaching consequences and applications in computer science, coding, and public-key cryptography.

Last, but not least, a strong grounding in arithmetic sets a student up for success in algebra. The division algorithm uses multiplication and subtraction. As such, division demands that we synthesise a lot of prior knowledge. This is what makes division challenging, and for many students it is their first taste of multi-layered processes. The ability to reflect on what you know, and implement it within a new, higher-level process is one of the generic mathematical skills that division helps to develop. The implementation of the division algorithm is typically a multi- step process, and as such it helps to develop skills that are invaluable when students move on to algebra. The link to factors is also critical in later years. return to top History The product of two numbers is the same no matter how you calculate it or how you write your answer. Just as the history of number is really all about the development of numerals, the history of multiplication and division is mainly the history of the processes people have used to perform calculations. The development of the Hindu-Arabic place-value notation enabled the implementation of efficient algorithms for arithmetic and was probably the main reason for the popularity and fast adoption of the notation.

The earliest recorded example of a division implemented algorithmically is a Sunzi division dating from 400AD in China. Essentially the same process reappeared in the book of al Kwarizmi in 825AD and the modern-day equivalent is known as Galley division. It is, in essence, equivalent to modern-day long division. However, it is a wonderful example of how notation can make an enormous difference. Galley division is hard to follow and leaves the page a mess compared to the modern layout. The layout of the long division algorithm varies between cultures. Throughout history there have been many different methods to solve problems involving multiplication. Some of them are still in use in different parts of the world and are of interest to teachers and students as alternative strategies or because of the mathematical challenge involved in learning them. return to top Italian or lattice method Another technique, known as the Italian or lattice method is essentially an implementation of the extended version of the standard algorithm but in a different layout. The method is very old and might have been the one widely adopted if it had not been difficult to print. It appears to have first appeared in India, but soon appeared in works by the Chinese and by the Arabs.

From the Arabs it found its way across to Italy and can be found many Italian manuscripts of the 14th and 15th centuries. The multiplication 34 × 27 is illustrated here. 34 × 27 = 918 In the top right rectangle 4 × 2 is calculated. The digit 8 is placed in the bottom triangle and 0 in the top triangle. Then 3 × 2 is calculated and the result entered as shown. In the bottom right rectangle 4 × 7 is calculated. The digit 8 is placed in the bottom triangle and the digit 2 in the top triangle. The result of 3 × 7 is also recorded in this way. The green diagonal contains the units. The blue diagonal contains the tens. The orange diagonal contains the hundreds. The digits are now summed along each diagonal starting from the right and each result recorded as shown. Note that there is a ‘carry’ from the ‘tens diagonal’ to the ‘hundreds diagonal’ 56 ÷ 8 = 7 and 56 ÷ 7 = 8. c 81 = 9 × 9 and 81 ÷ 9 = 9.

Because the divisor and the quotient are the same, the multiplication statement becomes a statement about squaring, and the other corresponding division statement is the same as the original statement. Exercise 4 a 24÷ 4 = 6. If 24 children are divided into groups of 4, how many groups are there? b 20 ÷ 2 = 10. If 20 metres of fabric is divided into two equal pieces, how long is c 160 ÷ 10 =16. If 160 books are bundled into packages of 10 each, how many packages are there? d 35÷7 = 5. If a 35-day period is divided into 7 equal periods, how long is each period? Exercise 5 a Twelve 7-person rescue teams can be formed, with 6 people to spare. How many people will be in 7 equal groups formed from 90 people? There will be 12 people in each group, with 6 left over. b Nine 5-seater cars are needed, and there will be two spare seats. How many people will be in 5 equal groups formed from 43 people?

There will be 8 groups, with 3 people left over. Exercise 6 Combined Probability question : GMAT Quantitative Section Combined Probability question : GMAT Quantitative Section Unanswered GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here . It appears that you are browsing the GMAT Club forum unregistered! Signing up is free, quick, and confidential. Join other 500,000 members and get the full benefits of GMAT Club Registration gives you: Tests Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT, Knewton, and others. All are free for GMAT Club members.

Applicant Stats View detailed applicant stats such as GPA, GMAT score, work experience, location, application status, and more Books/Downloads Download thousands of study notes, question collections, GMAT Club’s Grammar and Math books. All are free! and many more benefits! Hi GMATClubber! Thank you for using the timer! We noticed you are actually not timing your practice. Click the START button first next time you use the timer. There are many benefits to timing your practice, including: We’ll give you CLICK HERE TO HIDE/SHOW EVENTS CLICK HERE TO HIDE/SHOW EVENTS Schools: University of Toronto, Mcgill, Queens Followers: 0 Kudos [ ? ]: 92 [ 14 ] , given: 9 Combined Probability question [ #permalink ] BOOKMARKED Hi All, There are lots of probability questions which I found in this forum which worth practicing once you have them in one place. I copied them from this forum and all the credit goes to the original posters. Useful website to learn the simple probability. 1. Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

A: 1/5 E: 1/2 #p654985 2. Bill has a small deck of 12 playing cards made up of only 2 suits of 6 cards each. Each of the 6 cards within a suit has a different value from 1 to 6; thus, there are 2 cards in the deck that have the same value. Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value? A 8/33 E 25/33 56530.html 3. A man chooses an outfit from 3 different shirts, 2 different pairs of shoes, and 3 different pants. If he randomly selects 1 shirt, 1 pair of shoes, and 1 pair of pants each morning for 3 days, what is the probability that he wears the same pair of shoes each day, but that no other piece of clothing is repeated? 1) 5) snappy-dresser-p-86872.html 4.

In the xy- plane, a triangle has vertexes (0,0), (4,0) and (4,5). If a point (x,y) is selected at random from the triangular region, What is the probability that x-y>0 ? A. 1/5 E. 4/5 [url]what-is-the-probabilty-such-that-x-y0-86926.html[/url] 5. Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart." If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart? (A) 1/4 6. Laura has a deck of standard playing cards with 13 of the 52 cards designated as a "heart." If Laura shuffles the deck thoroughly and then deals 10 cards off the top of the deck, what is the probability that the 10th card dealt is a heart? (A) 1/4 7 .

At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples? # \frac{1}{12} 8 . A committee of three people is chosen from four married couples. What is the number of different committees that can be chosen if two people who are married to each other cannot both serve on the committee? 1. 16 9 . A fair coin is tossed 10 times. What is the probability that two heads do not occur consecutively? A) 1/ 2^4 10 . A drawer contains 8 socks, and 2 socks are selected at random without replacement. What is the probability that both socks are black? (1) The probability is less than 0.2 that the first sock is black. (2) The probability is more than 0.8 that the first sock is white. 11. Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address.

If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address? A) 1/24 12. A team has record of 12 wins and 13 losses for the season. Three games remain and if the probability of winning each remaining game is ½ and there are no draws. What is the probability that team finish the season with the winning record? 1. 1/8 13. Suppose you play a game where you role a single dice and whatever you roll you get that dollar amount. For example, if you roll a 1 you get $1. If you roll a 6 you get $6. If you are unhappy with the first roll, you can roll again. However, if you get lower the second time, you cannot take the first roll. If you are unhappy with the second roll you can roll a third and final time. Again, if you get lower on the third roll, you have to keep this roll and cannot take the first or second roll. What is the expected value of this game? 14 .

If a certain coin is flipped, the probability that the coin will land heads is 1/2. If the coin is flipped 5 times, what is the probability that it will land heads up on the first 3 flips and not on the last 2 flips? A. 3/5 16. Say you have 5 coins. 1 coin has heads on both sides and the other 4 coins are normal (heads on one side, tails on the other) A coin is selected at random and flipped five times, each time landing on heads. What is the probability that this coin is the coin that has heads on each side? Edit: the URL doesn't work, so I removed it 17. In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings? 5/21 22. There are 5 white balls and 3 red balls in a Jar. What is the probability- 1) if someone picks 2 balls without replacing, that 1 ball is white and the other is red?

2) what is the probability that the first ball is red and the second is white without replacing? 3) if someone picks 2 balls with replacing, that 1 ball is white and the other is red? 4) what is the probability that the first ball is red and the second is white with replacing? 23. Rich has 3 green, 2 red and 3 blue balls in a bag. He randomly picks 5 from the bag WITH replacement. What is the probability that of the 5 drawn balls, Rich has picked 1 red, 2 green, and 2 blue balls? 24. In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present? a. (13^4) x 48 x 47 b. (13^4) x 27 x 47 c. 48C6 31. The oasis output of Abu Ilan in the heart of the Negev desert, has a population of 20 Bedouin tribesmen and 20 Farma tribesmen.

El Kamin, a nearby oasis, has a population of 32 Bedouin tribesmen and 8 Farima tribesmen. A lost soldier (from another country), accidentally seperated from his army unit, is wandering through the desert and arrives at the edge of one of the oases. The soldier has no idea which oasis he has found, but the first person he spots from a distance is a Bedouin. What is the probability that he wandered into Abu Illan? What is the probability that he is in El Kamin? 38. Suppose you're given the choice of three boxes. In one of them is the big prize (full-ride from Stanford/Harvard/Any you like); in the others, an apples. The prize and the apples were placed randomly in the boxes. The rules of the game are as follows: After you have chosen a box, the box remains closed for the time being. Then one of the two remaining boxes is opened, (by the computer which knows what is in each box) and the box contains an apple (as far as there are two boxes with apples it's always possible to open one no matter what is in the box you chose).

After one box with apple is opened, you are asked to decide whether you want to stay with your first choice or to switch to the last remaining box. A. What is the probability of winning the prize if you stay with your first choice? B. What is the probability of winning the prize if you switch to the last remaining box? Basically question is: is it to your advantage to change your choice? 39. Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter she prepared an envelope with its correct address. If the 4 letters are to be put in 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address? And seems that it was confusing for many. GMAT often has similar questions, so find below the problems to master yourself in them. Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter she prepared an envelope with its correct address. If the 4 letters are to be put in 4 envelopes at random, what is the probability: A. That no letter will be put into the envelope with its correct address? B.

That all letters will be put into the envelope with its correct address? C. That only 1 letter will be put into the envelope with its correct address? D. That only 2 letters will be put into the envelope with its correct address? E. That only 3 letters will be put into the envelope with its correct address? F. That more than one letter will be put into the envelope with its correct address? G. That more than two letters will be put into the envelope with its correct address? 40. In the poker game Five-Card Draw, each player is dealt a hand consisting of 5 cards from a deck of 52 cards. Each card in the deck has a suit (clubs, hearts, diamonds, or spades) and a value (A, 2,..., 10,J,Q,K). A. What is the probability that we are dealt a four-of-a-kind? B. What is the probability that we are dealt full house? (A full house is a hand with both a three-of-a-kind and a two-of-a-kind.) C. What is the probability that we are dealt two pairs? (Fifth is different) D. What is the probability that we are dealt three of a kind?

(Others are different) E. What is the probability that we are dealt one pair? (Others are different) F. What is the probability that we are dealt all five of different ranks? G. What is the probability that we are dealt hands with every suit? 41. In a jar there are balls in different colors: blue, red, green and yellow. The probability of drawing a blue ball is 1/8. The probability of drawing a red ball is 1/5. The probability of drawing a green ball is 1/10. If a jar cannot contain more than 50 balls, how many yellow balls are in the Jar? a) 23. Challenge Question 57. A lottery game works as follows: The player draws a numbered ball at random from an urn containing five balls numbered 1, 2, 3, 4, and 5. If the number on the ball is even, the player loses the game and receives no points; if the number on the ball is odd, the player receives the number of points indicated on the ball. Afterward, he or she replaces the ball in the urn and draws again.

On each subsequent turn, the player loses the game if the total of the numbers becomes even, and gets another turn (after receiving the number of points indicated on the ball and then replacing the ball in the urn) each time the total remains odd. (a) What is the probability that the player loses the game on the third turn? Kudos [ ? ]: 158 [ 0 ], given: 0 Re: Combined Probability question [ #permalink ] Show Tags 29 Apr 2014, 02:42 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Question: If five baskets contain respectively 7, 6, 5, 4 and 3 apples, what is the average number of apples in two baskets?
Answer: Ten