Chapter 1 Use Positive Rational Numbers
Section 1.7: Solve Problems With Rational Numbers
Page 49 Exercise 1a Answer
To answer the question, we first need to calculate how much food the cat eats in one day.
Since Jenna feeds her cat twice a day, each time a \(\frac{3}{4}\) can, find the product of those two numbers.
Result
To answer the question, we first need to calculate how much food the cat eats in one day by multiplying 2 and \(\frac{3}{4}\).
Page 49 Exercise 1b Answer
Since Jenna feeds her cat twice a day, each time a \(\frac{3}{4}\) can, find teh product of those two numbers.
\(2 \times \frac{3}{4}=\frac{2}{1} \times \frac{3}{4}=\frac{2 \times 3}{1 \times 4}=\frac{6}{4}=\frac{4}{4}+\frac{2}{4}=1 \frac{1}{2}\)Jenna’s cat eats \(1 \frac{1}{2}\) can of food per day. Since Jenna’s friend will take car of her cat for 5 days multiply \(1 \frac{1}{2}\) by 5.
\(1 \frac{1}{2} \times 5=\frac{3}{2} \times 5=\frac{3}{2} \times \frac{5}{1}=\frac{3 \times 5}{2 \times 1}=\frac{15}{2}=\frac{14}{2}+\frac{1}{2}=7 \frac{1}{2}\)Jenna’s cat will eat \(7 \frac{1}{2}\) cans of food in five days, so since Jenna bought 8 cans of food she bought enough.
Result
Since Jenna feeds her cat twice a day, each time a \(\frac{3}{4}\) can, we can find the product of those two numbers to find how much the cat eats per day. We can then multiply the product by 5 to find how much food the cat eats in 5 days. This gives \(7 \frac{1}{2}\)cans which is less than 8 so she bought enough.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 49 Exercise 1 Answer
We know Jenna feeds her cat twice a day, feeds her cat \(\frac{3}{4}\) can each time, needs to buy enough for 5 days, and she bought 8 cans
If Jenna multiplied, divided, and compared to find out whether she has enough cat food, then Jenna first multiplied to find how many cans per day her cat eats:
\(2 \times \frac{3}{4}=\frac{2}{1} \times \frac{3}{4}=\frac{2 \times 3}{1 \times 4}=\frac{6}{4}=\frac{3}{2}\)Then she divided the number of cans she bought by the number of cans needed per day:
\(8 \div \frac{3}{2}=\frac{8}{1} \times \frac{2}{3}=\frac{8 \times 2}{1 \times 3}=\frac{16}{3}=5 \frac{1}{3}\)Jenna then bought enough cans for \(5 \frac{1}{3}\) days. Comparing this to 5 days, she can then conclude that she bought enough cans.
Result
Jenna multiplied 2 and \(\frac{3}{4}\) to find the number of cans needed per day. Next she divided the number of cans she bought by the number of cans needed per day. She then compared the quotient of \(5 \frac{1}{3}\) with 5 to determine that she bought enough.
Page 50 Exercise 1 Answer
A farmer is building a small horse-riding arena that is built using three rows of wood planks. The farmer uses wood planks that are \(8 \frac{1}{2}\) feet long and ordered 130 of these wood planks.
We need to determine if the farmer ordered enough planks for an arena that measured \(115 \frac{1}{4}\) ft by \(63 \frac{1}{2}\) ft.
To answer the question we must find the perimeter of the arena.
\(P=2 \times 115 \frac{1}{4}+2 \times 63 \frac{1}{2}=2 \times \frac{461}{4}+2 \times \frac{127}{2}=\frac{461}{2}+127\) \(=\frac{461}{2}+\frac{127 \times 2}{2}=\frac{461}{2}+\frac{254}{2}=\frac{461+254}{2}=\frac{715}{2}=357 \frac{1}{2}\)The farmer needs enough wood planks for 3 times the perimeter.
\(3 \times 357 \frac{1}{2}=\frac{3}{1} \times \frac{715}{2}=\frac{3 \times 715}{1 \times 2}=\frac{2145}{2}=1072 \frac{1}{2} \mathrm{ft}\)Divide to find how many \(8 \frac{1}{2}\)-foot-long planks are needed.
\(1072 \frac{1}{2} \div 8 \frac{1}{2}=\frac{2145}{2} \div \frac{17}{2}=\frac{2145}{2} \times \frac{2}{17}=\frac{2145 \times 2}{2 \times 17}=\frac{2145}{17}=126 \frac{3}{2}\)The farmer needs at least 127 wood planks to build the fencing. Since he ordered 130 wood planks, he has enough to build the fencing.
Result
The farmer is correct.
Page 51 Exercise 2 Answer
A 26.2-mile marathon is being planned.
The number of runners who finish the marathon is 320. Runners donate $2.50 for each mile they run. How much money is donated?
26.2 × 320 = 8384miles
320 runners finished the marathon which means they ran all 26.2 miles. Thus, together they ran 8384 miles. For each mile they donated $2.50.
8384 × $2.50 = 20960
Result
$20960 is donated.
Page 52 Exercise 1 Answer
There are two types of problems with rational numbers – problems with fractions and problems with decimals.
When solving multistep problems with fractions or decimals, first decide the steps to use to solve the problem and choose the correct operations. Then identify the information you need from the problem and use it correctly. Calculate and interpret solutions and check that the answer is reasonable.
Result
When solving multistep problems with fractions or decimals, first decide the steps to use to solve the problem and choose the correct operations. Then identify the information you need from the problem and use it correctly. Calculate and interpret solutions and check that the answer is reasonable.
Page 52 Exercise 2 Answer
Meghan has \(5 \frac{1}{4}\) yards of fabric and plans to use \(\frac{2}{3}\) of the fabric to make 4 identical backpacks. Meghan multiplies \(5 \frac{1}{4}\) by \(\frac{2}{3}\) to find how much fabric she will use to make the backpacks:
\(5 \frac{1}{4} \times \frac{2}{3}=\frac{21}{4} \times \frac{2}{3}=\frac{21 \times 2}{4 \times 3}=\frac{42}{12}=\frac{36}{12}+\frac{6}{12}=3 \frac{6}{12}=3 \frac{1}{2}\)Meghan will use \(3 \frac{1}{2}\) yards of fabric to make 4 backpacks.
To find out how much fabric she needs for each backpack she must divide \(3 \frac{1}{2}\) by 4.
\(3 \frac{1}{2} \div 4=\frac{7}{2} \div \frac{4}{1}=\frac{7}{2} \times \frac{1}{4}=\frac{7 \times 1}{2 \times 4}=\frac{7}{8}\)Meghan needs \(\frac{7}{8}\) yard of fabric for each backpack.
Result
She must divide \(3 \frac{1}{2}\) by 4 to get \(\frac{7}{8}\) which means she needs \(\frac{7}{8}\) yard of fabric for each backpack.
Page 52 Exercise 3 Answer
Each side of a square patio is 10.5 feet. The patio is made up of 1.5 foot by 1.5 foot square stones. What is the number of stones in the patio?
The solution which is given:
10.5 x 10.5 = 110.25, 110.25 ÷ 1.5 = 73.5
To solve the problem we need to divide the area of the patio by the area of the stones. In the solution, the area of the patio is correctly calculated, 10.5 x 10.5 = 110.25. However, the area is then divided by 1.5 which is not the area of the stone, it is the length of the side of the stone. To find the solution we first need to find the area of the stone.
The area of the stone is 2.25 square feet.
To calculate how many stones are in the patio we must divide the area of the patio, 110.25, by the area of the stones.
There are 49 stones in the patio.
Result
The solution does not include all the steps needed to solve the problem. The find the solution, we need to find the area of the stone. There are 49 stones in the patio.
Page 52 Exercise 4a Answer
Devon records 4 hours of reality shows on her DVR and records comedy shows for \(\frac{3}{8}\) of that amount of time.
To find the number of hours of comedy shows that Devon records, we need to multiply 4 by \(\frac{3}{8}\).
\(\frac{3}{8} \times 4=\frac{3}{8} \times \frac{4}{1}=\frac{3 \times 4}{8 \times 1}=\frac{12}{8}=\frac{3}{2}=1 \frac{1}{2}\)Result
Devon recorded \(1 \frac{1}{2}\) hour of comedy shows.
Page 52 Exercise 4b Answer
Devon records 4 hours of reality shows on her DVR and \(1 \frac{1}{2}\) hours of comedy shows, which we calculated in Exercise 4a.
To find the total number of hours of reality and comedy shows that Devon records, we need to add the number of hours of reality and the number of hours of comedy shows.
\(4+1 \frac{1}{2}=\frac{4}{1}+\frac{3}{2}=\frac{8}{2}+\frac{3}{2}=\frac{8+3}{2}=\frac{11}{2}=\frac{10}{2}+\frac{1}{2}=5 \frac{1}{2}\)Result
The total number of hours of reality and comedy shows that Devon recored is \(5 \frac{1}{2}\).
Page 52 Exercise 4c Answer
The total number of hours of reality and comedy shows that Devon recorded is \(5 \frac{1}{2}\), which we calculated in Exercise 4b. Devon watches all the reality and comedy shows in half-hour sittings.
To find the number of half-hour sittings needed to watch all the shows we must divide the total number of hours of reality and comedy shows that Devon recorded, which is \(5 \frac{1}{2}\), by \(\frac{1}{2}\).
\(5 \frac{1}{2} \div \frac{1}{2}=\frac{11}{2} \div \frac{1}{2}=\frac{11}{2} \times \frac{2}{1}=\frac{11 \times 2}{2 \times 1}=\frac{22}{2}=11\)Result
The number of half-hour sittings needed to watch all the shows is 11.
Page 52 Exercise 5a Answer
An auto mechanic earns $498.75 in 35 hours during the week, his pay is $2.50 more per hour on weekends, and he works 6 hours on the weekend in addition to 35 hours during the week. We need to find how much he earns.
The questions we need to answer to solve the problem are:
1. How much does he earn per hour during the week?
2. How much does he earn per hour during the weekend?
3. How much does he earn during the whole weekend?
Page 52 Exercise 5b Answer
The questions we need to answer to solve the problem are:
1. How much does he earn per hour during the week?
2. How much does he earn per hour during the weekend?
3. How much does he earn during the whole weekend?
To answer the first question, we must divide $498.75 by 35, since he earns $498.75 during the whole week and he works 35 hours during the whole week.
During the week, the auto mechanic earns $14.25 per hour.
To answer the second question, we must add $2.50 to $14.25, since he earns $2.50 more per hour on weekends.
During the weekend, the auto mechanic earns $16.75 per hour. To answer the third question, we must multiply $16.75 by 6, since he earns $16.75 per hour and works 6 hours during the whole weekend.
During the whole weekend, the auto mechanic earns $100.50.
To determine how much the auto mechanic earns, we must add how much he earns during the whole week and how much he earns during the weekend.
Result
The auto mechanic earns $599.25.
Page 53 Exercise 6 Answer
First, we must calculate how much we would pay separately for apples, pears, and oranges we bought.
The dollar amounts must be rounded to the nearest cent so the oranges cost $2.79, the pears cost $1.49, and the apples cost $3.14.
Second, we add the cost of apples, pears, and oranges to get a total bill.
Result
Total bill rounded to the nearest cent is $7.42.
Page 53 Exercise 7a Answer
First, to solve the problem, we must calculate how much 8.9 pounds of apples cost. Since apples cost $0.99 per lb we multiply 8.9 and 0.99.
8.9 x 0.99 = $8.81
8.9 pounds of apples cost $8.81.
Result
First, we must calculate how much 8.9 pounds of apples cost, which is $8.81.
Page 53 Exercise 7b Answer
First, to solve the problem, we must calculate how much does 8.9 pounds of apples cost, which we did in Exercise 7a.
8.9 pounds of apples cost $8.81.
Next, we have to calculate the difference between the money the student payed with and the cost of 8.9 pounds of apples.
10.00 – 8.81 = 1.19
The student receives $1.19 of change.
Result
Next, we have to calculate the difference between the money the student paid with and the cost of the 8.9 pounds of apples. The student receives $1.19 in change.
Page 53 Exercise 8a Answer
To solve the problem we must first calculate how many oranges and how many pears the customer bought. Since he paid $3.27 for oranges and they cost $1.09 lb, we must divide 3.27 by 1.09. We use the same logic to find how many pears he bought, so we divide 4.76 by 1.19.
The customer bought 3 pounds of oranges and 4 pounds of pears.
Result
We must first calculate how many oranges and how many pears the customer bought. The customer bought 3 pounds of oranges and 4 pounds of pears.
Page 53 Exercise 8b Answer
To solve the problem we must first calculate how many oranges and how many pears the customer bought, which we did in Exercise 8a.
The customer bought 3 pounds of oranges and 4 pounds of pears.
To find how many pounds of fruit the customer bought we now must add the number of pounds of oranges and the number of pounds of pears he bought.
3 + 4 = 7
The customer bought 7 pounds of fruit.
Result
Next, we must add the number of pounds of oranges and the number of pounds of pears he bought. The customer bought 7 pounds of fruit.
Page 53 Exercise 9 Answer
It is given that students put \(2 \frac{1}{4}\) pounds of trail mix into bags that each weigh \(\frac{3}{8}\) pound and bring \(\frac{2}{3}\) of the bags of trail mix on a hiking trip.
To find how many bags are left, we would need to perform the following steps:
Find how many bags the students made by dividing \(2 \frac{1}{4}\) pounds by \(\frac{3}{8}\).
Multiply the number of bags by \(\frac{2}{3}\) to find how many they brought on the trip.
Subtract the total number of bags and the number of bags they brought to find how many are left.
We would then need 3 steps so no we cannot complete just one step to determine how many bags are left.
Performing step 1 we get:
\(2 \frac{1}{4} \div \frac{3}{8}=\frac{9}{4} \times \frac{8}{3}=\frac{72}{12}=6\)Performing step 1 gives:
\(6 \times \frac{2}{3}=\frac{12}{3}=4\)Performing step 3 then gives:
6 − 4 = 2
There are then 2 bags left.
Result
No, we cannot complete just one step. We need to find how many bags they made, how many they took on the hike, and then we can find how many are left, which was 2 bags.
Page 53 Exercise 10 Answer
It is given that \(\frac{3}{5}\) of the T-shirts are blue and \(\frac{5}{8}\) of the blue shirts are on sale. The fraction of T-shirts that are blue and on sale is then:
\(\frac{3}{5} \times \frac{5}{8}=\frac{3 \times 5}{5 \times 8}=\frac{15}{40}=\frac{3}{8}\)If \(\frac{1}{3}\) of the blue shirts that are on sale are size medium, then the fraction of blue T-shirts that are medium and on sale is:
\(\frac{1}{3} \times \frac{3}{8}=\frac{1 \times 3}{3 \times 8}=\frac{3}{24}=\frac{1}{8}\)Result
\(\frac{1}{8}\)Page 54 Exercise 11 Answer
The area of the community garden is given as the product 6.2 × 4.5 and the area of the vegetable garden is 0.4 times the area of the community garden.
6.2 x 4.5
0.4 x 27.9
Result
The area of the vegetable garden is 11.16 square meters.
Page 54 Exercise 12 Answer
The area of the herb garden is given by the product 2.2 x 1.6 and the area of the flower garden is 9.7 square meters greater than the herb garden.
Result
The area of the flower garden is 13.22 square meters.
Page 54 Exercise 13 Answer
It is given that \(\frac{3}{4}\) cup is left and that Jim will divide \(\frac{4}{5}\) of the leftover dip equally between 2 friends.
Since \(\frac{3}{4}\) cup is left and he divides up \(\frac{4}{5}\) of the leftover dip, then the amount of dip he divides between his friends is:
\(\frac{3}{4} \times \frac{4}{5}=\frac{3 \times 4}{4 \times 5}=\frac{12}{20}=\frac{3}{5}\)Since he divides it equally between 2 friends, then each friend gets half of the \(\frac{3}{5}\) cup of dip:
\(\frac{1}{2} \times \frac{3}{5}=\frac{1 \times 3}{2 \times 5}=\frac{3}{10}\)Each friend will then get \(\frac{3}{10}\) cup.
Result
\(\frac{3}{10}\) cup
Page 54 Exercise 14 Answer
On the first day of a 3 day hiking trip the students hike 0.28 of the total distance which is 18.5 kilometers.
On the first day the students hiked 5.18 kilometers.
The remaining distance is equally split between the second and the third day.
The remaining distance is 13.32 kilometers. It is divided by two.
Result
On day 3 they will hike 6.66 kilometers.
Page 54 Exercise 15 Answer
From the picture, we can see that the three containers of potato salad that Kelly bought are different weights 1.03 lb, 1.12 lb, and 1.6 lb. First, we must calculate the sum 1.03 + 1.12 + 1.6 to find how many pounds she bought and then multiply that by \(\frac{4}{5}\) to find how many she brought to the picnic.
\((1.03+1.12+1.6) \times \frac{4}{5}=3.75 \times \frac{4}{5}=3.75 \times 0.8=3\)
Another way to find the answer is to write 3.75 as a fraction and than multiply that fraction by \(\frac{4}{5}\).
\(3.75=\frac{375}{100}=\frac{15}{4}\) \(3.75 \times \frac{4}{5}=\frac{15}{4} \times \frac{4}{5}=\frac{15 \times 4}{4 \times 5}=\frac{60}{20}=3\)Result
Kelly brought 3 pounds of potato salad to the picnic.
Page 54 Exercise 16 Answer
To calculate how much did the students earn, we must first calculate how man bottles they sold. 84.5 ounces od liquid soap was divided into 6.5 ounce bottles.
The students sold bottles of soap for $5.50 and they sold 13 of them.
Result
The students earned $71.50.
Page 54 Exercise 17 Answer
Last week it took Clair \(\frac{7}{12}\) hour to mow one lawn.
\(\frac{7}{12} \times 5=\frac{7}{12} \times \frac{5}{1}=\frac{7 \times 5}{12 \times 1}=\frac{35}{12}=\frac{24}{12}+\frac{11}{12}=2 \frac{11}{12}\)Last week it took her \(2 \frac{11}{12}\) to mow 5 lawns. This week she mowed the same lawns but it took her \(\frac{5}{12}\) hour because she used her new lawn mower.
\(\frac{5}{12} \times 5=\frac{5}{12} \times \frac{5}{1}=\frac{5 \times 5}{12 \times 1}=\frac{25}{12}=\frac{24}{12}+\frac{1}{12}=2 \frac{1}{12}\)This week it took Claire \(2 \frac{1}{12}\) hour to mow 5 lawns.
\(2 \frac{11}{12} \div 2 \frac{1}{12}=\frac{35}{12} \div \frac{25}{12}=\frac{35}{12} \times \frac{12}{25}=\frac{35 \times 12}{12 \times 25}=\frac{35}{25}=\frac{7}{5}=\frac{5}{5}+\frac{2}{5}=1 \frac{2}{5}\)Result
Claire’s time to mow all the lawns last week was \(1 \frac{2}{5}\) times longer than this week.