ACT Math - Proportion / Ratio / Rate

1

The largest of three numbers is six less than four times the smallest number. The middle number is three more than twice the smallest number. The sum of twice the largest number, three more than three times the middle number, and four times the smallest number is 90. What is the value of the largest number?

14

10

12

18

22

Explanation

Because all the numbers are expressed in terms of the smallest number, let that number be represented by x. Therefore, the largest number can be expressed as 4x-6, and the middle number can be expressed as 2x+3. Plug those numbers into the addition equation and solve for x. Then plug the value for x into the expression representing the largest number.

2(4x-6) + (3+3(2x+3)) + 4(x) = 90 --> 18x = 90 --> x = 5

4(6)-6 = 14

2

A new car can travel an average of 63 miles per gallon of gasoline. Gasoline costs $5.05 per gallon. How much would it cost to travel 6,363 miles in this car?

$505

$101

$405.05

$510

$510.05

Explanation

First, find the total amount of gas necessary for the trip. 6363/63 = 101 gallons (easy to see as 63 * 100 = 6300 + 1 * 63 = 6363). Then multiply the number of gallons by the price per gallon of gasoline, 5.05 * 101 = $510.05 and is your answer (again, easy to see when 5.05 * 100 + 1 * 5.05).

3

The largest of three numbers is six less than four times the smallest number. The middle number is three more than twice the smallest number. The sum of twice the largest number, three more than three times the middle number, and four times the smallest number is 90. What is the value of the largest number?

14

10

12

18

22

Explanation

Because all the numbers are expressed in terms of the smallest number, let that number be represented by x. Therefore, the largest number can be expressed as 4x-6, and the middle number can be expressed as 2x+3. Plug those numbers into the addition equation and solve for x. Then plug the value for x into the expression representing the largest number.

2(4x-6) + (3+3(2x+3)) + 4(x) = 90 --> 18x = 90 --> x = 5

4(6)-6 = 14

4

A new car can travel an average of 63 miles per gallon of gasoline. Gasoline costs $5.05 per gallon. How much would it cost to travel 6,363 miles in this car?

$505

$101

$405.05

$510

$510.05

Explanation

First, find the total amount of gas necessary for the trip. 6363/63 = 101 gallons (easy to see as 63 * 100 = 6300 + 1 * 63 = 6363). Then multiply the number of gallons by the price per gallon of gasoline, 5.05 * 101 = $510.05 and is your answer (again, easy to see when 5.05 * 100 + 1 * 5.05).

5

At the beginning of a race, a person's speed is $\small 5$ miles per hour. One hour into the race, a person increases his speed by $\small 50%$ . A half an hour later, he increases again by another $\small 25%$ . If he finishes this race in two hours, what is the average speed for the entire race? Round to the nearest hundredth of a mile per hour.

$\small 6.72\ mph$

$\small 7.29\ mph$

$\small 5.67\ mph$

$\small 7.03\ mph$

$\small 6.45\ mph$

Explanation

Recall that in general $\small R=\frac{D}{T}$

Now, let's gather our three rates:

Rate 1: $\small 5$

Rate 2: $\small 5*1.5 = 7.5$

Rate 3: $\small 7.5 * 1.25=9.375$

Now, we know that the time is a total of $\small 2$ hours. Based on our data, we can write:

$\small R=\frac{5*1 + 7.5 * 0.5 + 9.375 * 0.5}{2}=\frac{13.4375}{2}$

This is $\small 6.71875$ miles per hour, which rounds to $\small 6.72$ .

6

At the beginning of a race, a person's speed is $\small 5$ miles per hour. One hour into the race, a person increases his speed by $\small 50%$ . A half an hour later, he increases again by another $\small 25%$ . If he finishes this race in two hours, what is the average speed for the entire race? Round to the nearest hundredth of a mile per hour.

$\small 6.72\ mph$

$\small 7.29\ mph$

$\small 5.67\ mph$

$\small 7.03\ mph$

$\small 6.45\ mph$

Explanation

Recall that in general $\small R=\frac{D}{T}$

Now, let's gather our three rates:

Rate 1: $\small 5$

Rate 2: $\small 5*1.5 = 7.5$

Rate 3: $\small 7.5 * 1.25=9.375$

Now, we know that the time is a total of $\small 2$ hours. Based on our data, we can write:

$\small R=\frac{5*1 + 7.5 * 0.5 + 9.375 * 0.5}{2}=\frac{13.4375}{2}$

This is $\small 6.71875$ miles per hour, which rounds to $\small 6.72$ .

7

Twenty bakers make $\small 720$ dozen cookies in eight hours. How many cookies does each baker make in an hour?

$\small 54$

$\small 7$

$\small 1080$

$\small 90$

$\small 4.5$

Explanation

This problem is a variation on the standard equation $\small W=RT$ . The $\small R$ variable contains all twenty bakers, however, instead of just one. Still, let's start by substituting in our data:

$\small 720 = R * 8$

Solving for $\small R$ , we get $\small 90$ .

Now, this represents how many dozen cookies the whole group of $\small 20$ make per hour. We can find the individual rate by dividing $\small 90$ by $\small 20$ , which gives us $\small 4.5$ . Notice, however, that the question asks for the number of cookies—not the number of dozens. Therefore, you need to multiply $\small 4.5$ by $\small 12$ , which gives you $\small 54$ .

8

Twenty bakers make $\small 720$ dozen cookies in eight hours. How many cookies does each baker make in an hour?

$\small 54$

$\small 7$

$\small 1080$

$\small 90$

$\small 4.5$

Explanation

This problem is a variation on the standard equation $\small W=RT$ . The $\small R$ variable contains all twenty bakers, however, instead of just one. Still, let's start by substituting in our data:

$\small 720 = R * 8$

Solving for $\small R$ , we get $\small 90$ .

Now, this represents how many dozen cookies the whole group of $\small 20$ make per hour. We can find the individual rate by dividing $\small 90$ by $\small 20$ , which gives us $\small 4.5$ . Notice, however, that the question asks for the number of cookies—not the number of dozens. Therefore, you need to multiply $\small 4.5$ by $\small 12$ , which gives you $\small 54$ .

9

A group of 15 friends is having lunch together. Each person eats at least 2/3 of a pizza. What is the smallest number of whole pizzas needed for lunch?

10

5

15

7

Explanation

The minimum number of whole pizzas needed is 15(2/3) = 10.

10

A group of 15 friends is having lunch together. Each person eats at least 2/3 of a pizza. What is the smallest number of whole pizzas needed for lunch?

10

5

15

7

Explanation

The minimum number of whole pizzas needed is 15(2/3) = 10.

Proportion / Ratio / Rate

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation