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Solutions to the Fraction Word Problems 1. Susan has two and 1/2 hours to rehearse her speech. Her speech is 3 pages long. If she wants to spend the same amount of time on each page, how much time will she have to spend on each page? How many times does 3 go into 2 and 1/2? Since this is a division problem, we should change the mixed numbers to improper fractions. Now invert and multiply She should spend 5/6 of an hour on each page. 2. Melissa and Michelle went skiing at Mammoth. 1/3 of the mountain is rocky, and another 1/4 is snow covered. What fraction of the mountain is either rocky or snow covered? Add 1/3 + 1/4 Find common denominators. The smallest common denominator for 4ths and 3rds is 12. 3. In #2, the rest of the mountain is covered in forest. What fraction of the mountain is forested? The rest of the mountain is covered in forest so we take away the amount that is covered in rock or snow, and the rest is covered in forest. 1 - 7/12 Again we need common denominators. Change the 1 to 12/12 12/12 - 7/12 = 5/12 4. A recipe for pancakes calls for 3/4 cup of flour and Bob is making it two and a half times the number of pancakes called for in the recipe. How much flour will he need? This is an A = Rate x Base type of problem. The rate is 3/4 cup per recipe. Since the units of the rate are cups per recipe, the base is the number of recipes, so you multiply Since we are multiplying, change the mixed number to an improper fraction. 5. Jane's car gets Thirty and one third mpg. She bought four and one quarter gallons. How far can she travel before she runs out of gas? This is another A = RB type of problem. The mileage is 30 and 1/3 miles per gallon, so that is the rate. The word that comes after the "per" is the base. That is gallons. Since we have both the rate and the base, we multiply. Since we are multiplying mixed numbers, we should convert them to improper fractions Multiply tops and bottoms. Since this is an improper fraction, you may want to change it to a mixed number. And the answer is This is a fairly reasonable answer. If the car gets just over 30 mi/gal, and you buy a little more than 4 gal, you should be able to get a little farther than 120 miles. 6. Ten children went to soccer practice. 1/2 decided to go for pizza afterwards. It takes 3 children to eat 1 pizza. How many pizzas did they eat? This is a multi-part problem. First we have to find 1/2 of the children. "of" means multiply, so we multiply 10x1/2. The 2 in the bottom will cancel with the 10 on the top, and since 2 goes into 10 5 times, 1/2 of 10 children are 5 children. Now that we know how many children we have we can find out how many pizzas we need. It takes 3 children to eat 1 pizza. That gives us 3 children per pizza. Since we have the word "per' in the units, the 3 children per pizza is a rate. Since the word "pizza" comes after the word "per", the pizzas are the base. In the A = RB types of problems, if we need to find the base, the base is what we need to multiply times the rate to get the amount. The amount in this case is the 5 children, so we need to find out what we need to multiply times 3 children per pizza to get 5 children. To find out what you need to multiply, divide. This is the B = A/R type of problem. 5 children / 3 children per pizza = 5/3 pizzas. If we turn this into a mixed number it will give us 1 and 2/3 pizzas. 7. In # 6, they ordered 2 pizzas. After the kids ate their pizza how much was left over for the adults? Subtract 1 and 2/3 from 2 and you are left with 1/3 of a pizza for the adults. 8. If it takes 1/4 hr to type 3/4 of a report, how long does it take to type the entire report? This can be done as an A = RB type of problem. In this case the rate would be the number of hours per report. In that case, the base would be the number of reports, which, in this case is the fraction 3/4, and the amount would be the number of hours, which, in this case is the fraction 1/4. When you are looking for the rate you use the equation R = A/B which is Invert and multiply The 4's cancel and the answer is 1/3 hr. This can also be done using ratio and proportion. When you set up the ratio and proportion problem you have you will also find that you are going to divide 1/4 by 3/4. 9. It is 3/4 of a mile to school. One day when Jane walked to school, she got 2/3 of the way there before she remembered that she had forgotten her homework. She walked back home, got her homework, and walked to school. How far did she walk? The first question here is what is 2/3 of 3/4? "of" means multiply After canceling we are left with an answer of 1/2. 10. There is 8/9 of a mile from home to school. I have to walk 1/4 of the distance. How far is that? What is 1/4 of 8/9? 11. If Jane loses four and a half pounds per week, how long will it take her to lose 21 and a half pounds? 4 and 12 lb per week is a rate. The base is the number of weeks, and the amount is the number of pounds. If we are looking for the number of weeks, we use the formula B = A/R How many times does 4 and 1/2 go into 21 and 1/2? Since we are dividing mixed numbers, we will want to change them to improper fractions. 12. Clear Lake holds two hundred fifty and 9/10 acre feet of water. Lake Sonoma holds eighteen and 4/5 acre feet. How many Lake Sonomas will it take to fill Clear Lake? How many times will 18 and 4/5 go into 250 and 9/10? We are dividing mixed numbers so turn them into improper fractions. Invert and multiply The 5 will cancel with the 10, but we should decide if the 2509 will cancel with the 94. Well, 94 = 2x47, so the only primes that go into 94 are 2 and 47. 2 doesn't go into 2509 because 2509 is an odd number, and you can check to find out that 47 does not go into 2509 either, so they don't cancel. Our answer is then 2509/188. This is one instance where a mixed number would be more meaningful than an improper fraction. So there would be enough water in Clear Lake to fill up Lake Sonoma 13 and 55/188 times. 13. Linda weighs seventy-five and 3/5 pounds. Big Bobby weighs three and 1/3 times that. How much does Big Bobby weigh? What is 3 and 1/3 times 75 and 3/5? Change to improper fractions 14. There was 5/6 of a pie left. Mary and Joe ate 3/4 of that. How much pie did they eat? What is 3/4 of 5/6? 15. John gathered 12 eggs on Monday. On Tuesday, he gathered 1/2 of that amount. On Wednesday, he gathered 1/2 of the amount of eggs that he gathered on Tuesday. How many eggs did he gather on Wednesday? First we have to figure out how many eggs John gathered on Tuesday. 1/2 of 12 is 6. On Wednesday he gathered half of that, and 1/2 of 6 is 3. So altogether, John gathered 12 + 6 + 3 = 21 eggs. 16. Susan needed 2/6 of a pound of sugar for 1 batch of cookies. She made seven and a half batches of cookies. How much sugar did she use? The 2/6 lb per batch is a rate. We multiply the number of lb/batch by the number of batches to get the number of lbs. First notice that the 2/6 reduces to 1/3. It would be a good idea to reduce it before doing anything else. Next, since we are multiplying mixed numbers, change the mixed number to an improper fraction. If we change the 5/2 to a mixed number, she used 2 and 1/2 lb of sugar. 17. Ann spent 3/4 of an hour on the phone. She talked to Sally 1/2 of the time. How much time did she spend talking to Sally? What is 1/2 of ¾ 1/2 x 3/4 = 3/8 18. Mary had a jar of jelly beans 2/3 full. She divided them equally between herself and a friend. How much did each friend get? Mary divided her jelly beans into 2 equal pieces. Invewrt and multiply. The 2 to the right of the division sign has an invisible denominator of 1, so when you invert it you get 1/2 Each of them would get 1/3 of a jar of jelly beans 19. Sarah is on a diet. She can only eat 1/4 lb. of turkey breast for lunch. She stops by the deli and orders 1/4 lb of turkey breast. The clerk slices off 3 slices (all the same size) and weighs them. The three slices weigh a total of 1/3 lb. Sarah says, "I can't eat 1/3 lb. I'm on a diet." The clerk says, "Fine, I'll only charge you for 1/4 lb. Sarah says, "But I can't eat 1/3 lb. I'm on a diet." The clerk says, then just eat "1/4 lb. and give the rest to your dog." How many slices can Sarah eat? If we want to use the A = RB technique, we would need to find the number of slices per pound. We would then multiply that times the number of pounds which Sarah can eat, 1/4, to find the number of slices. If 3 slices weigh 1/3 lb the rate is We need to divide 3 by 1/3 to get the rate If there are 9 slices in a pound, there will be 1/4 of that in 1/4 lb, and 9 x 1/4 = 9/4 slices. We may want to express that as a mixed number. This problem can also be done using ratio and proportion. This gives rise to the following equation When we multiply both sides by 1/4 we get which is the same answer we got the other way. She should eat 2 and 1/4 slices of turkey breast if she wants to limit hser consumption of turkey breast to only 1/4 lb. 20. On a remote desert island a team of anthropologists finds that 2/3 of the adult females are married to 1/2 of the adult males. Assuming that all marriages are monogamous and heterosexual, what fraction of the adults on the island are married? A picture is helpful in dealling with this problem. We want to match up 2/3 of the females with 1/2 of the men Using fraction strips to solve this problem: The shaded regions represent the married adults, and the unshaded regions represent the unmarried adults. If we express the 1/2 of the men as 2/4 of the men, then the regions for the men and the women matcvh up, and we see that there are a total of 4 shaded regions out of 7 total regions. As a result, 4/7 of the adults are married. This is a problem where you would want to find common numerators and add the tops and bottoms. 21. Sallie Jo was mixing up a cheesecake. The recipe called for 1 package of cream cheese to go with 5 eggs along with some other ingredients. Unfortunately, Sallie Jo had only 3 eggs, so she had to adjust. How much of the package of cream cheese should she use? Again a picture would be helpful. If 5 eggs will go with 1 whole cheesecake, then 3 eggs will go with 3/5 of a cheesecake. 22. Sallie Jo was mixing up a cheesecake. This time she had a different recipe which said that she needed 2 packages of cream cheese to go with 9 eggs. Unfortunately, she only had 7 eggs. How much cream cheese should she use if she adjusts the recipe to use 7 eggs? This time the picture is good to use. We can see that we will be using 7/9 of 2 pounds of creamcheese or 14/9 which becvomes 1 and 5/9 when we turn it into a mixed number. This problem can be done using the A = RB technique. Evidently the proper rate of creamcheese to eggs is 2 pounds to 9 eggs or 2/9 pounds per egg. See the 2/9 of a pound which is associated with each egg. Then we multiply the 2/9 pound per egg by 7 eggs and get 14/9 pounds of creamcheese. This could also be done using ratio and proportion. 23. Jill was making fudge. The recipe called for 3/4 of a pound of chocolate, but Jill only had 2/5 of a pound of chocolate. What part of the recipe could Jill make? We can use the A= RB technique. In this case the most natural rate that occurs to us is 3/4 lb per recipe. In that case the base would be recipes, so if we are looking for the number of recipes we should use the formuls B = A/R where A is 2/5 lb and R is 3/4 lb/recipe. 24. Tom and Tina Termite were jogging when Tina said to Tom, "How far do you think we've gone, Tom?" We've been out here for 3/4 of an hour and we usually go 8/11 of a foot per hour." At that rate, how far would they have traveled? This is an A = RB type problem. They would have traveled 6/11 of a foot in 3/4 of an hour. 25. Someone had cut off part of the stick of butter, so Carol couldn't read it all, but the part that she could read was marked "6 tablespoons - 3/4 stick". How many tablespoons of butter are there in a whole stick? This can also be done as an A = RB type of problem. The rate here is the number of tablespoons per stick. Since we do not know the number of tablespoons per stick, we use the formula R = A/B, where, since the units are tablespoons per stick, we put the tablespoons on top and the sticks on the bottom.