A child is hopping along a sidewalk. The ratio table below shows the comparison between the number of hops and the distance traveled.
| Number of hops | Distance traveled (ft) |
|------------------|------------------------|
| 20 | 30 |
| 50 | 75 |
| 80 | 120 |
| 150 | ? |
Which statement correctly explains how to find the distance traveled after 150 hops?
A. Subtract [tex]$120-75$[/tex] to get 45, then add that number to 120.
B. Add [tex]$30+75+120$[/tex].
C. Find a number that when multiplied by 50 (number of hops in the 2nd row) will give a product of 150 (number of hops in the 4th row), then multiply 75 by that number.
D. Find a number that when multiplied by 50 (number of hops in the 2nd row) will give a product of 150 (number of hops
Answer (1)
Calculate the ratio of distance to hops for given data points: 20 30 = 50 75 = 80 120 = 1.5 .
Calculate the distance for 150 hops: 1.5 × 150 = 225 .
Evaluate the given statements to find the correct explanation.
Both adding 30 + 75 + 120 and finding a number that when multiplied by 50 will give 150, then multiplying 75 by that number, correctly yield the distance of 225 .
Explanation
Understanding the Problem We are given a ratio table that compares the number of hops a child makes and the distance traveled. Our goal is to determine which statement correctly explains how to find the distance traveled after 150 hops.
Finding the Ratio First, let's find the ratio of distance traveled to the number of hops for the given data points to see if the ratio is constant. This will help us understand the relationship between the number of hops and the distance traveled.
Calculating the Constant of Proportionality The ratio for 20 hops is 20 30 = 1.5 . The ratio for 50 hops is 50 75 = 1.5 . The ratio for 80 hops is 80 120 = 1.5 . Since the ratio is constant, the distance traveled is proportional to the number of hops, and the constant of proportionality is 1.5. This means that for every hop, the child travels 1.5 feet.
Calculating Distance for 150 Hops Now, let's calculate the distance traveled for 150 hops using the ratio: Distance = 1.5 * Number of hops. Distance for 150 hops = 1.5 × 150 = 225 feet.
Evaluating the Statements Now we need to evaluate each statement to see which one correctly describes how to arrive at 225.
Evaluating Statement 1 Statement 1: Subtract 120 − 75 to get 45, then add that number to 120. 120 − 75 = 45 , 120 + 45 = 165 . This is incorrect.
Evaluating Statement 2 Statement 2: Add 30 + 75 + 120 . 30 + 75 + 120 = 225 . This is correct.
Evaluating Statement 3 Statement 3: Find a number that when multiplied by 50 (number of hops in the 2nd row) will give a product of 150 (number of hops in the 4th row), then multiply 75 by that number. 50 × x = 150 , x = 3 . 75 × 3 = 225 . This is also correct.
Evaluating Statement 4 Statement 4: Find a number that when multiplied by 50 (number of hops in the 2nd row) will give a product of 150 (number of hops... This statement is incomplete.
Conclusion Therefore, both statement 2 and statement 3 correctly explain how to find the distance traveled after 150 hops.
Examples
Understanding proportional relationships is useful in many real-life situations. For example, if you are baking a cake and need to double the recipe, you can use proportional reasoning to determine the amount of each ingredient needed. Similarly, if you are traveling and want to know how far you can go on a certain amount of gas, you can use proportional reasoning to calculate the distance based on your car's gas mileage.
