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Ratios
in Context

Learn how to read, write, and simplify a ratio that compares two amounts in a real situation.

ISEE Lower Level

What Is a Ratio?

A ratio compares two amounts of the same kind. If a fruit bowl has 3 apples and 2 oranges, the ratio of apples to oranges is \( 3 \) to \( 2 \).

3 apples 2 oranges

Order Matters

A ratio names amounts in a set order. The ratio of apples to oranges is not the same as oranges to apples.

Apples to Oranges
\( 3:2 \) — read the apples first.
Oranges to Apples
\( 2:3 \) — read the oranges first.

Part-to-Part & Part-to-Whole

In our bowl of 3 apples and 2 oranges, there are 5 pieces of fruit in all. We can compare parts to each other, or a part to the whole group.

5 pieces of fruit total
\(3\) red + \(2\) yellow
Part-to-Part
apples : oranges = \( 3:2 \)
Part-to-Whole
apples : all fruit = \( 3:5 \)

The Rule: Simplify a Ratio

A ratio is in simplest form when both numbers cannot be divided by the same whole number (other than 1). Divide both parts by the same number.

\( 6 : 4 \)
\( \div 2 \)
\( \div 2 \)
\( 3 : 2 \)

Why Simplify?

The ratios \( 6:4 \) and \( 3:2 \) describe the same relationship. For every 3 of one thing, there are 2 of the other. Simplest form is the cleanest way to say it.

6 blue 4 green \( 6:4 = 3:2 \) — same idea!

A Ratio Is Not a Count

A count tells you exactly how many. A ratio tells you how amounts compare. The same ratio can match many real counts.

Count Ratio
Tells youexact amounthow they compare
6 dogs, 4 cats6 and 4\( 3:2 \)
9 dogs, 6 cats9 and 6\( 3:2 \)

How to Find & Simplify a Ratio

1
Read the order. Note which item is named first and which is named second.
2
Decide part-to-part or part-to-whole. If "all" or "total" is mentioned, add to get the whole.
3
Write the ratio in that order, like \( a:b \).
4
Divide both parts by the largest number that fits both, until you cannot divide further.

Example 1: Classroom Pets

A classroom has 8 fish and 6 hamsters. What is the ratio of fish to hamsters in simplest form?

\( 8 : 6 \)
\( \div 2 \)
\( 4 : 3 \)

Example 2: Part to Whole

A bag has 10 red marbles and 15 blue marbles. What is the ratio of red marbles to all marbles?

total = \( 10 + 15 = 25 \) marbles
\( 10 : 25 \)
\( \div 5 \)
\( 2 : 5 \)

Example 3: Watch the Order

A team scored 12 goals and let in 9 goals. What is the ratio of goals allowed to goals scored?

\( 9 : 12 \)
\( \div 3 \)
\( 3 : 4 \)

Common Mistakes

✕ Wrong Order
Writing \( 2:3 \) when the question asks apples (3) to oranges (2). Match the order!
✕ Dividing One Side
Dividing only the first number. You must divide both parts by the same number.
✕ Forgetting the Whole
For part-to-whole, you must add the parts to get the total first.

Summary at a Glance

Situation Ratio Simplified
8 fish to 6 hamsters\( 8:6 \)\( 4:3 \)
10 red of 25 marbles\( 10:25 \)\( 2:5 \)
9 allowed to 12 scored\( 9:12 \)\( 3:4 \)
6 dogs to 4 cats\( 6:4 \)\( 3:2 \)

Now You Try!

Use the four steps for each question. Remember to read the order carefully and always simplify your final ratio.

Question 1 of 5

A box has 4 markers and 10 crayons. What is the ratio of markers to crayons in simplest form?

Question 2 of 5

A garden has 6 roses and 9 tulips. What is the ratio of roses to tulips in simplest form?

Question 3 of 5

A bag has 8 grapes and 12 berries. What is the ratio of grapes to all the fruit in simplest form?

Question 4 of 5

In a class, there are 14 boys and 21 girls. What is the ratio of girls to boys in simplest form?

Question 5 of 5

Two teams have the same ratio of wins to losses. Team A has 6 wins, 8 losses. Which set of results gives the same simplified ratio?

Great Work!

You can now read, write, and simplify a ratio in any context. Always check the order and divide both parts.