Writing Ratios from Word Problems | Grade 6 Math | LevelKite

You already know how to write two kinds of ratio. A ratio compares two quantities. A quantity is a number together with what it counts or measures, such as $5$ red marbles or $2$ cups of water. A part-to-part ratio compares one part of a group with another part. A part-to-whole ratio compares one part with the whole group. In this lesson you will write each ratio with a colon, such as $5:3$ , and read it as "5 to 3".

Now you will find ratios inside word problems. In a word problem, a story describes a situation, and then a question asks for one ratio. Your job is to work out which two quantities the question wants, and in which order.

The same story can lead to several different ratios. Here is one story. A bag holds only red marbles and blue marbles. There are $5$ red marbles and $3$ blue marbles. The table shows four different questions about this bag. Watch how the ratio changes with the question.

Four questions about a bag that holds only 5 red marbles and 3 blue marbles

The question asks for the ratio of	First quantity	Second quantity	Ratio
red marbles to blue marbles	$5$ red marbles	$3$ blue marbles	$5:3$
blue marbles to red marbles	$3$ blue marbles	$5$ red marbles	$3:5$
red marbles to all the marbles	$5$ red marbles	$5 + 3 = 8$ marbles in all	$5:8$
blue marbles to all the marbles	$3$ blue marbles	$5 + 3 = 8$ marbles in all	$3:8$

The story stayed the same, but the ratio changed each time. Two things made the difference.

The order. The first two rows use the same two numbers. The ratio of red marbles to blue marbles is $5:3$ , because there are $5$ red marbles for every $3$ blue marbles. The ratio of blue marbles to red marbles is $3:5$ . These are two different ratios. Writing $3:5$ for red to blue would say the bag has fewer red marbles than blue marbles, and that is false.
The quantities. The last two rows compare a part with the whole bag. The story never states the number $8$ , so you have to work it out. The bag holds only red and blue marbles, so you add the two parts to get the whole: $5 + 3 = 8$ .

Key concept

The question chooses the ratio

The story describes the situation. The question chooses which two quantities to compare, and in which order. When a question asks for the ratio of one quantity to another, the first quantity it names gives the first number, and the second quantity it names gives the second number.

\text{first quantity named} : \text{second quantity named}

To write a ratio from a word problem, take three steps:

Name. Find the two quantities the question names, and note their order.
Number. Find the number for each quantity. If the story does not give it, work it out from the numbers the story does give.
Order. Write the two numbers in the question's order, with a colon between them.

The three examples below follow these steps.

Worked example

Both numbers are in the story

A garden bed has $8$ tomato plants and $5$ pepper plants. What is the ratio of pepper plants to tomato plants?

Name. The question names pepper plants before the word "to" and tomato plants after it. So the number of pepper plants goes first and the number of tomato plants goes second.
$\text{pepper plants} : \text{tomato plants}$
Number. The story gives both numbers: there are 5 pepper plants and 8 tomato plants. Nothing needs to be worked out.
Order. Write the numbers in the question's order, with pepper plants first. The story lists tomato plants first, but the question sets the order, so 5 goes first.
$5 : 8$

Answer

The ratio of pepper plants to tomato plants is $5:8$ .

Worked example

You have to find the whole

You make a fruit drink by mixing $5$ cups of orange juice with $2$ cups of water. Nothing else goes in. What is the ratio of cups of water to cups of fruit drink?

Name. The question names the water first and the fruit drink second. The water is one part of the drink and the fruit drink is the whole, so this is a part-to-whole ratio. Every amount here is in cups.
$\text{cups of water} : \text{cups of fruit drink}$
Number. The story gives the amount of water: 2 cups. It does not give the amount of fruit drink. The drink contains only orange juice and water, and both amounts are in cups, so add the two parts to find the whole.
$5 + 2 = 7$
Order. Write the numbers in the question's order: 2 cups of water first, then 7 cups of fruit drink.
$2 : 7$

Answer

The ratio of cups of water to cups of fruit drink is $2:7$ .

Here is a check that helps you spot that mistake. Read your ratio back with labels. Say each number with the thing it counts or measures, then compare with the question.

The mistaken ratio $2:5$ reads "2 cups of water to 5 cups of orange juice". The question asked for water to fruit drink, so the second label does not match.
The correct ratio $2:7$ reads "2 cups of water to 7 cups of fruit drink". Both labels match the question, in the same order.

This check helps you spot a wrong quantity or a wrong order. It does not check your arithmetic, so check any adding or subtracting on its own.

Worked example

You have to find a part

A team played $15$ games this season. It won $8$ games and lost all the others. What is the ratio of losses to wins?

Name. The question names losses first and wins second. It does not name all the games, so the total, 15, is not one of the two quantities. You still need 15 to find a missing number.
$\text{number of losses} : \text{number of wins}$
Number. The story gives the number of wins: 8. It does not give the number of losses. Every game was a win or a loss, so the number of wins plus the number of losses equals the 15 games played.
$8 + \text{number of losses} = 15$
Number, continued. Subtraction undoes addition. If 8 plus the number of losses makes 15, then the number of losses is 15 minus 8. Check by adding back: 8 + 7 = 15.
$\text{number of losses} = 15 - 8 = 7$
Order. Write the numbers in the question's order: 7 losses first, then 8 wins.
$7 : 8$
Check. Read the ratio back with labels: 7 losses to 8 wins. The question asked for losses to wins, so the labels match, in the same order.

Answer

The ratio of losses to wins is $7:8$ .

Takeaway. In a ratio word problem, the story describes the situation and the question chooses the comparison. Name the two quantities the question asks about. Find the number for each one, and work it out if the story does not give it. Write the numbers in the question's order, then read your ratio back with labels. Next, you will use tape diagrams, drawings made of equal-sized boxes, to represent a ratio.