Find Mistakes in Written Ratios | Grade 6 Math | LevelKite

A ratio compares two quantities, which are amounts or counts of things. It tells you how much there is of one thing compared with another. You can write a ratio with the word to or with a colon: $4$ to $9$ , or $4:9$ .

In this lesson you are the checker. You will look at ratios that someone else wrote. For each one, you will decide whether it is right. If it is not, you will say exactly what went wrong.

Picture a bag that holds $4$ red tiles and $9$ blue tiles, and nothing else. The ratio of red tiles to blue tiles is $4:9$ . It says there are $4$ red tiles for every $9$ blue tiles.

So each number in a ratio has a job. In this lesson, every number in a ratio is an actual count of something in the situation. In a correct ratio, the first number counts the quantity named first, and the second number counts the quantity named second. Here is the ratio of red tiles to blue tiles with each number labeled by what it counts.

\underbrace{4}_{\text{red tiles}}:\underbrace{9}_{\text{blue tiles}}

Key concept

Read the ratio back into words

To check a ratio, say what each number counts. That turns the ratio back into words. Then compare your words with the words of the question, and run two checks:

Right quantities? Do the two numbers count the two quantities the question names?
Right order? Does the quantity the question names first come first in the ratio?

A ratio written with actual counts is correct when both answers are yes. If either answer is no, you have found a mistake.

The two checks give the two kinds of mistake you will meet in this lesson. A ratio with the right quantities in the wrong order has an order mistake. A ratio in which a number counts the wrong thing has a quantity mistake.

One common quantity mistake is to mix up a part and the whole. A part is one group inside a collection. The whole is the entire collection. In the bag, the red tiles are one part and the blue tiles are the other part. The bag holds nothing else, so the whole is all the tiles: $4 + 9 = 13$ . A part-to-part ratio compares one part with another part. A part-to-whole ratio compares one part with the whole.

Now watch the two checks at work. A question asks for the ratio of red tiles to all tiles in the bag. Four students wrote the answers in the table. Read each written ratio back into words yourself. Then compare your words with the table.

Question: the ratio of red tiles to all tiles. The bag has 4 red, 9 blue, 13 in all.

Written ratio	What it compares	Verdict
$4:9$	red tiles to blue tiles	Quantity mistake. The $9$ counts blue tiles, a part. The question names all the tiles, the whole.
$13:4$	all tiles to red tiles	Order mistake. The quantities are right, but red tiles must come first.
$9:13$	blue tiles to all tiles	Quantity mistake. The $9$ counts blue tiles, but the question names red tiles.
$4:13$	red tiles to all tiles	Correct. Right quantities in the right order.

Worked example

Check Mia's ratio

A shelter has $7$ cats and $5$ dogs. A question asks for the ratio of dogs to cats. Mia wrote $7:5$ . What mistake did Mia make, and what is the correct ratio?

Start with the question. It names dogs first and cats second. So a correct ratio has the number of dogs first and the number of cats second.
$\text{dogs}:\text{cats}$
Say what each of Mia's numbers counts. The shelter has 7 cats, so her first number counts cats. It has 5 dogs, so her second number counts dogs. Read back into words, her ratio compares cats to dogs.
$\underbrace{7}_{\text{cats}}:\underbrace{5}_{\text{dogs}}$
Run the two checks. Right quantities? Yes, because cats and dogs are the two quantities the question names. Right order? No, because the question puts dogs first and Mia put cats first. So this is an order mistake.
Fix the ratio by giving each count the place the question gives it. Put the 5 dogs first and the 7 cats second.
$5:7$

Answer

Mia made an order mistake. Her ratio $7:5$ compares cats to dogs. The ratio of dogs to cats is $5:7$ .

Worked example

Check Leo's ratio

A basket holds only apples and pears: $8$ apples and $3$ pears. A question asks for the ratio of apples to all the fruit. Leo wrote $8:3$ . What mistake did Leo make, and what is the correct ratio?

Start with the question. It names apples first and all the fruit second. Apples are a part and all the fruit is the whole, so the question asks for a part-to-whole ratio.
$\text{apples}:\text{all the fruit}$
Say what each of Leo's numbers counts. His 8 counts apples, which matches the quantity named first. His 3 counts pears. Pears are the other part, not the whole. So Leo's ratio does not have the right quantities. This is a quantity mistake.
$\underbrace{8}_{\text{apples}}:\underbrace{3}_{\text{pears}}$
Find the number that belongs in second place. The basket holds only apples and pears, so the two parts together make the whole. Add them to count all the fruit.
$8 + 3 = 11$
Write the ratio the question asks for. Keep the 8 apples first, because that place was already right. Put the 11 pieces of fruit second.
$8:11$

Answer

Leo made a quantity mistake. He wrote $8:3$ , the part-to-part ratio of apples to pears. The ratio of apples to all the fruit is $8:11$ .

Worked example

Check Dev's ratio

A class has $25$ students. Of these, $11$ walk to school and the rest ride the bus. A question asks for the ratio of bus riders to walkers. Dev wrote $25:11$ . What mistake did Dev make, and what is the correct ratio?

Compare the question with Dev's numbers. The question names bus riders first and walkers second, and both are parts of the class. Dev's 11 counts walkers, which matches. His 25 counts all the students. That is the whole, not the bus riders, so this is a quantity mistake.
$\underbrace{25}_{\text{all students}}:\underbrace{11}_{\text{walkers}}$
Find the part that belongs in first place. Every student either walks or rides the bus, so walkers and bus riders together make 25. Subtract the 11 walkers from 25 to leave the bus riders. Check by adding back: 14 + 11 = 25.
$25 - 11 = 14$
Write the ratio the question asks for, with the 14 bus riders first and the 11 walkers second.
$14:11$

Answer

Dev made a quantity mistake. He used the whole class, $25$ , where the question names the bus riders. The ratio of bus riders to walkers is $14:11$ .

Takeaway. A ratio is more than two numbers. It is right only when it compares the quantities the question names, in the order the question names them. To check a ratio, read it back into words by saying what each number counts. If the right quantities are in the wrong order, it is an order mistake. If a number counts the wrong thing, such as a part where the whole belongs, it is a quantity mistake.