Lesson Using proportions to solve word problems

Using proportions to solve word problems

This lesson is the continuation of the lesson Proportions of this module.
Here you can see some typical word problems that can be solved using proportions.

Problem 1. Cost of apples at a store

3 pounds of apples cost 2 dollars 40 cents at a store.
How much it will cost to buy 5 pounds of apples at the store if no discount is applied?

Solution
Let us denote as the cost in cents to buy 5 pounds of apples at the store.
If 3 pounds of apples cost 2 dollars 40 cents (240 cents), then one pound costs cents.
You can express the same price of one pound in another way as the ratio , where the numerator is the unknown cost of 5 pounds of apples.
Since it is the same price, you can write the equality of these two ratios
.
This is the proportion with the unknown mean term.
Apply the rule for solving proportions from the lesson Proportions: the unknown mean of the proportion is equal to the product of extremes divided by the known mean.
You get the unknown cost for 5 pounds of apples
cents.

Answer. The cost of 5 pounds of apples at the store is 4 dollars.

Problem 2. Cost of copies at the Copy Center

60 copies at the Copy Center cost 9 dollars.
How much it will cost to make 270 copies at the Copy Center at the same rate?

Solution
Let us denote as the cost in cents of 270 copies at the Copy Center.
You can express the price of one copy as the ratio dividing the cost of 60 copies by the number of copies.
You can express the same price in another way as the ratio of the unknown cost of 270 copies to that number of copies.
Since both ratios represent the same number, you can write the equality of these ratios
.
This is the proportion with the unknown mean term.
Apply the rule for solving proportions from the lesson Proportions: the unknown mean of the proportion is equal to the product of extremes divided by the known mean.
You get the unknown cost for 270 copies
cents.

Answer. The cost of 270 copies at the Copy Center is 40 dollars 50 cents.

Problem 3. Distance traveled by a car

A car traveled 130 miles at a constant speed moving for two hours along a highway.
What is the distance traveled by the car if it was moving for 4.5 hours at the same speed?

Solution 1
Let us denote as the distance traveled by the car if it was moving for 4.5 hours at the same speed.
You can express the speed of the car as the ratio where the numerator is the traveled distance, and the denominator is the time spent.
You can express the speed of the car in another way as the ratio dividing the unknown distance by the time spent of 4.5 hours.
Since both ratios represent the same speed, you can write the equality of these ratios:
.
This is the proportion with the unknown extreme term.
Use the rule for solving proportions from the lesson Proportions: the unknown extreme of the proportion is equal to the product of means divided by the known extreme.
Applying this rule you get the unknown distance
miles.

Answer. The distance traveled by the car is equal to 292.5 miles if it was moving for 4.5 hours with the same speed.

Solution 2
You can solve the same problem differently.
First, calculate the speed of the car. To do this, divide the distance traveled by the car during two hours (130 miles) by the time spent (2 hours).
You will get miles per hour as the value of the car speed.
Now, calculate the distance traveled by the car for 4.5 hours.
To do this, simply multiply the car speed, 65 miles per hour, by the time 4.5 hours.
You get miles, the same answer.

Problem 4. Time of the trip

The car traveled 130 miles along a highway moving for two hours at a constant speed.
What time is needed for the car to travel 227.5 miles along the highway if it was moving at the same speed?

Solution 1
Let us denote as the time in hours, which car needed to travel 227.5 miles.
Similar to Problem 1, you can express the speed of the car as the ratio of the traveled distance of 130 miles to the time spent of 2 hours.
You can express the speed of the car in another way as the ratio dividing the distance of miles by the unknown time .
Since both ratios represent the same speed, you can write the equality of these ratios:
.
This is the proportion with the unknown mean term.
Use the rule for solving proportions from the lesson Proportions: the unknown mean of the proportion is equal to the product of extremes divided by the known mean.
Applying it you get the unknown time
hours.

Answer. The time required for the car to travel 227.5 miles is 3.5 hours if it was moving with the same speed.

Solution 2
You can solve the same problem differently.
First, calculate the speed of the car. This was done in the Solution 2 to Problem 1.
You get miles per hour for the car speed.
Now, calculate the time required for the car to make 227.5 miles.
To do this, simply divide the distance of 227.5 miles to the car speed, 65 miles per hour.
You get hours, the same answer.

Problem 5. Fuel amount a car consumes for the trip

The consumed fuel rate depends on the car model and on the number of other conditions such as the car speed, the road slope and so on.
A car consumed two gallons of gasoline to travel 65 miles moving along a highway at a constant speed.
What amount of the gasoline is needed for the car to travel 227.5 miles moving under the same conditions?

Solution
Let us denote as the gasoline amount (in gallons), which is needed for the car to travel 292.5 miles.
You can express the distance the car moves when it spends one gallon of the gasoline as the ratio dividing the distance of 65 miles by the spent gasoline amount of 2 gallons.
You can express the same distance in another way as the ratio dividing the distance value of miles by the unknown amount of gasoline .
Since the driving conditions are the same, both these ratios represent the same number.
Therefore you can write the equality of these ratios:
.
This is the proportion with the unknown mean term.
Apply the rule for solving proportions from the lesson Proportions: the unknown mean of the proportion is equal to the product of extremes divided by the known mean.
You get the unknown gasoline amount
gallons.

Answer. The amount of the gasoline is 7 gallons for the car trip of 227.5 miles.