Envision Math Accelerated Grade 7 Volume 1 Student Edition Chapter 3 Analyze And Use Proportional Relationships
Question. Basketball contest, Elizabeth made 9 out of 25 free throw attempts. Alex made 8 out of 20 free throw attempts. Janie said that Elizabeth had a better free-throw record because she made more free throws than Alex. Find out whether Janie’s reasoning is correct or not.
Given: In a basketball contest, Elizabeth made 9 out of 25 free throw attempts.
Alex made8 out of 20 free throw attempts. Janie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
We need to find out whether Janie’s reasoning is correct or not.
Finding the rates of each of them:
For Elizabeth.
\(\frac{\text { Number of throws Made }}{\text { Number of attempts }}=\frac{9}{25}\)= 0.36
For Alex
\(\frac{\text { Number of throws Made }}{\text { Number of attempts }}=\frac{8}{20}\)= 0.4
The rate Alex is more. Therefore, Alex had a better free-throw record because she made more free throws than Elizabeth.
Janie’s reasoning is wrong.
Given: In a basketball contest, Elizabeth made 9 out of 25 free throw attempts.
Alex made8 out of 20 free throw attempts.
Janie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
We need to decide who had the better free-throw record. Justify your reasoning using mathematical arguments.
Finding the rates of each of them:
For Elizabeth:
\(\frac{\text { Number of throws Made }}{\text { Number of attempts }}=\frac{9}{25}\)= 0.36
For Alex:
\(\frac{\text { Number of throws Made }}{\text { Number of attempts }}=\frac{8}{20}\)= 0.4
The rate of Alex is more. Therefore, Alex had a better free-throw record because she made more free throws than Elizabeth.
Alex had a better free-throw record because she made more free throws than Elizabeth.
Question. Explain which mathematical model we used to justify our reasoning.
We need to explain which mathematical model we used to justify our reasoning.
Also, explain if there are other models you could use to represent the situation.
Finding the rates of each of them.
The rate for Elizabeth is 0.36
The rate for Elizabeth is 0.4
The rate of Alex is more.
Therefore, Alex had a better free-throw record because she made more free throws than Elizabeth.
The mathematical model which I used here is finding the rates.
We could also use ratios and unit rates to determine the same.
The mathematical model which I used here is finding the rates. We could also use ratios and unit rates to determine the same.
Question. Explain how are ratios, rates, and unit rates used to solve problems.
We need to explain how are ratios, rates, and unit rates used to solve problems.
A rate is nothing but a measurement between two different quantities.
A ratio is a measurement between two same quantities.
A unit rate is a rate at which one quantity differs per unit of another quantity.
Examples are:
Rate –
Distance \(=\frac{\text { Speed }}{\text { Time }}\)
Here, speed and time are of different quantities. Ratio – \(\frac{x}{y}\)
Unit rate –
Distance \(=\frac{50 \text { miles }}{2 \text { hour }}\)
= 25 miles per hour
Here, 25 miles per hour is the unit rate.
The ratios, rates, and unit rates are used to solve problems in order to compare varying quantities by determining the number of units of one quantity per one unit of another quantity.
Question. Jennifer is a lifeguard at the same pool. She earns $137.25 for 15 hours of lifeguarding. Determine how much Jennifer earns per hour.
Given that, Jennifer is a lifeguard at the same pool. She earns $137.25 for 15 hours of lifeguarding.
We need to determine how much Jennifer earns per hour.
Finding the unit rate of Jennifer, we get
\(\frac{\text { Amount in dollars }}{\text { Number of hours }}=\frac{137.25}{15}\)= 9.15 dollars per hour
Jennifer earns $9.15 per hour.
Question. A kitchen sink faucet streams 0.5 gallons of water in 10 seconds. A bathroom sink faucet streams 0.75 gallons of water in 18 seconds. Find which faucet will fill a 3-gallon container faster.
Given that, A kitchen sink faucet streams 0.5 gallons of water in 10 seconds.
A bathroom sink faucet streams 0.75 gallons of water in 18 seconds.
We need to find which faucet will fill a 3−gallon container faster.
The rate of kitchen sink faucet is:
\(\frac{\text { Number of gallons of water }}{\text { Number of seconds }}=\frac{0.5}{10}\)
= 0.05 gallons of water per second.
The rate of bathroom sink faucet is:
\(\frac{\text { Number of gallons of water }}{\text { Number of seconds }}=\frac{0.75}{18}\)
= 0.042 gallons of water per second
The rate of bathroom sink faucet is more. This means that the bathroom sink faucet fills faster.
The bathroom sink faucet will fill a 3−gallon container faster.
Question. Explain how are ratios, rates, and unit rates used to solve problems.
We need to explain how are ratios, rates, and unit rates used to solve problems.
A rate is nothing but a measurement between two different quantities.
A ratio is a measurement between two same quantities.
A unit rate is a rate at which one quantity differs per unit of another quantity.
Examples are:
Rate –
Distance \(=\frac{\text { Speed }}{\text { Time }}\)
Here, speed and time are of different quantities. Ratio – \(\frac{x}{y}\)
Unit rate –
Distance \(=\frac{50 \text { miles }}{2 \text { hour }}\)
= 25 miles per hour
Here, 25 miles per hour is the unit rate.
The ratios, rates, and unit rates are used to solve problems in order to compare varying quantities by determining the number of units of one quantity per one unit of another quantity.
Question. Dorian buys 2 pounds of almonds for $21.98 and 3 pounds of dried apricots for $26.25. Which is less expensive per pound? How much less expensive?
Given:
Dorian buys 2 pounds of almonds for $21.98 and 3 pounds of dried apricots for $26.25.
To find/solve:
Which is less expensive per pound? How much less expensive?
Almonds:
\(\frac{21.98}{2}\)= \(\frac{21.98 \div 2}{2 \div 2}\)
= \(\frac{10.99}{1}\)
Apricots:
\(\frac{26.25}{3}\)= \(\frac{26.25 \div 3}{3 \div 3}\)
= \(\frac{8.75}{1}\)
Now equivalent ratios we get, 10.99 – 8.75 = 2.24, Apricots for 2.24
Apricots for 2.24.
Question. Krystal is comparing two internet service plans. Plan 1 cost $34.99 per month. Plan 2 costs $134.97 every 3 months. Find the krystal plans to stay with one service plan for 1 year, which should she choose? How much will she save?
Given:
Krystal is comparing two internet service plans.
Plan 1 cost $34.99 per month. Plan 2 costs $134.97 every 3 months.
To find/solve:
If Krystal plans to stay with one service plan for 1 year, which should she choose? How much will she save?
Plan 1 :
\(\frac{34.99}{1}\)= \(\frac{34.99 .12}{1.12}\)
= \(\frac{419.88}{12}\)
Plan 2 :
\(\frac{134.97}{1}\)= \(\frac{134.97 \div 3}{3 \div 3}\)
= \(\frac{44.99}{1}\)
Now finding equivalent ratios. Plan 1 is cheaper for 539.88 – 419.88 = 120.
In plan 1 she will save.
Question. Pam read 126 pages of her summer reading book in 3 hours. Zack read 180 pages of his summer reading book in 4 hours. Find they continue to read at the same speeds, will they both finish the 215 page book after 5 total hours of reading?
Given:
Pam read 126 pages of her summer reading book in 3 hours.
Zack read 180 pages of his summer reading book in 4 hours.
To find/solve:
If they continue to read at the same speeds, will they both finish the 215-page book after 5 total hours of reading?
Pam:
\(\frac{126}{3}\)=\(\frac{126 \div 3}{3 \div 3}\)
\(=\frac{42}{1}\) \(=\frac{42.5}{1.5}=\frac{210}{5}\)Zack:
\(\frac{180}{4}\)=\(\frac{180 \div 4}{4 \div 4}\)
\(=\frac{45}{1}\) \(=\frac{45.5}{1.5}=\frac{225}{5}\)She reads 42 pages per hour and he reads 45 pages per hour. Hence only Zack will finish.
Zack will finish.
Question. Megan walked 5 miles; her activity tracker had counted 9,780 steps. David’s activity tracker had counted 11,928 steps after he walked 6 miles. Find the steps for 1 mile.
Given:
Megan walked 5 miles; her activity tracker had counted 9,780 steps. David’s activity tracker had counted 11,928 steps after he walked 6 miles.
We know that:
Megan’s steps:
We find the steps for 1 mile:
Steps \(=\frac{9780}{5}\)
Steps = 1956
Megan takes 1956 steps per mile.
David’s steps:
We find the steps for 1 mile:
Steps \(=\frac{11928}{6}\)
Steps = 1988
David walks 1988 steps per mile.
Therefore David takes more steps to walk a mile.
1988−1956 = 32 more steps to walk a mile.
David takes more steps to walk 1 mile. David takes 32 more steps.
Question. A package of 5 pairs of insulated gloves costs $29.45. Find the price for 1 glove.
A package of 5 pairs of insulated gloves costs $29.45.
We know that:
We find the price for 1 glove:
Price \(=\frac{29.45}{5}\)
Price = 5.89
A single pair of gloves cost $ 5.89.
Question. Yellow packet white rice costs $6.30 for 18 ounces, Blue packet white rice costs $4.46 for 12 ounces and the green packet of white rice costs $2.59 for 7 ounces. Find the price for 1 ounce of the yellow packet of white rice.
Given:
Yellow packet white rice costs $6.30 for 18 ounces.
Blue packet white rice costs $4.56 for 12 ounces.
The green packet of white rice costs $2.59 for 7 ounces.
We know that:
We find the price for 1 ounce of the yellow packet of white rice:
Price \(=\frac{6.30}{18}\)
Price = 0.35
The price for 1 ounce of Yellow packet rice is $0.35.
We find the price for 1 ounce of the blue packet of white rice:
Price \(=\frac{4.56}{12}\)
Price = 0.38
The price for 1 ounce of Blue packet rice is $0.38.
We find the price for 1 ounce of the green packet of white rice:
Price \(=\frac{2.57}{7}\)
Price = 0.36
The price for 1 ounce of Green packet rice is $0.36.
The yellow package has the lowest cost per ounce of rice.
Question. The 5 panes cost $14.25. She breaks 2 more panes while repairing the damage. Find the cost for 1 pane of glass.
Given:
The 5 panes cost $14.25.
She breaks 2 more panes while repairing the damage.
We find the cost for 1 pane of glass:
Cost \( = \frac{14.25}{5}\)
Cost = 2.85
The cost for 1 pane of glass is $2.85.
We find the cost of 2 panes of glass:
Cost = 2.85 × 2
Cost = 5.7
The cost for 2 panes of glass is $5.70.
Question. The fare for 36 miles is $25.20. Find the cost per mile.
Given:
The fare for 36 miles is $25.20.
We first find the cost per mile:
Cost \(=\frac{25.20}{36}\)
Cost = 0.7
The cost per mile is 25.20.
Therefore, the required fare is
\(=\frac{25.20}{36}\)\(=\frac{x}{47}\)
\(x=\frac{25.20 \times 47}{36}\)x = 33.92
The fare for a 47-mile ride is $33.92
Therefore, the cost per mile is $0.7 and the fare for 47 miles is $33.92.
Question. Company A has 12 tigers for $33.24. Company B has 6 tigers for $44.80 and Company C 15 tigers for $41.10. Find the cost per tiger for all the companies.
Given:
Company A: 12 tigers for $33.24.
Company B: 16 tigers for $44.80.
Company C: 15 tigers for $41.10.
We first find the cost per tiger for all the companies:
Company A
Cost = \(\frac{33.24}{12}\)
Cost = 2.77
The cost per tiger for Company A is $2.77.
Company B
Cost = \(\frac{44.80}{16}\)
Cost = 2.8
The cost per tiger for Company B is $2.80.
Company C
Cost = \(\frac{41.10}{15}\)
Cost = 2.74
The cost per tiger for Company C is $2.74.
Company C has the lowest cost per tiger.
Therefore, Company C has the lowest cost per tiger.
Given:
Company C 15 tigers for $41.10.
The cost per tiger for Company C is $2.74.
The students plan to sell the tigers for $5 each.
We know that the lowest cost per tiger is $2.74 for company C.
The students sell the mascots according to $5.
Therefore the profit that the students will make for each tiger they sell is:
profit = 5−2.74
profit = 2.26
Therefore, the students will obtain the profit of $2.26 for each tiger they sell.
Question. A contractor purchases 7 dozen pairs of padded work gloves for $103.32. He incorrectly calculates the unit price at $14.76 per pair. Find the cost of 1 unit pair of padded work gloves.
Given:
A contractor purchases 7 dozen pairs of padded work gloves for $103.32.
He incorrectly calculates the unit price at $14.76 per pair.
To find the cost of 1 unit pair of padded work gloves, we divide $103.32 by 84
Cost = \(\frac{103.32}{84}\)
Cost = 1.23
Therefore, the cost of 1 unit pair of padded work gloves is $1.23.
Therefore, the unit price is $1.23.
Given:
A contractor purchases 7 dozen pairs of padded work gloves for $103.32.
He incorrectly calculates the unit price at $14.76 per pair.
To find the cost of 1 unit pair of padded work gloves, we divide $103.32 by 84
Cost = \(\frac{103.32}{84}\)
Cost = 1.23
Therefore, the cost of 1 unit pair of padded work gloves is $1.23.
He should have divided the total price by 84 since he bought 7 dozen pairs of gloves and 1 dozen equals to 12 units.
Therefore, the error the contractor was likely to make is total number of units. He should remember that there are 84 units and 7 dozen pair of gloves.
Question. A warehouse store sells 5.5-ounce cans of tuna in packages of 6 that cost $9.24. The store also sells 6.5 ounces cans of the same tuna in packages of 3 cans for $4.68. Find the cost of 1 ounce for each package.
Given:
A warehouse store sells 5.5-ounce cans of tuna in packages of 6 that cost $9.24.
The store also sells 6.5 ounces cans of the same tuna in packages of 3 cans for $4.68.
It also sells 3.5-ounce cans in packages of 4 cans for $4.48.
We find the cost of 1 ounce for each package:
Package 1: Cost = 9.24/(5.5 × 5) = 9.24/27.5 = 0.33
Package 2: Cost = 4.68/(6.5 × 3) = 4.68/19.5 = 0.24
Package 3: Cost = 4.48/(3.5 × 4) = 4.48/14 = 0.32
We see that package 3 has the lowest cost per ounce of tuna that is $0.32.
Therefore, package 3 which cost $0.32 per ounce has the lowest cost per ounce of tuna.
Question. Irene’s car had 6 gallons of gas in its 15-gallon tank. Irene wants to fill it at least halfway. The gas costs $3.80 per gallon. Find the total gallon gas has filled.
Given:
Irene’s car had 6 gallons of gas in its 15-gallon tank.
Irene wants to fill it at least halfway. The gas costs $3.80 per gallon.
Irene’s car has a capacity of 15 gallons and it already has 6 gallons of gas filled.
Therefore the remaining capacity of the car’s tank is 9 gallons.
Irene wants to fill the tank at least halfway.
The half of the tank is 7.5 gallons.
We need 1.5 gallons to fill the tank halfway.
If Irene fill her tank it would cost her 9×3.80=34.20$
Therefore, Irene does not need more than 1.5 gallons of gas, and if Irene adds $3.80 and $7.60 worth