Ratio and Proportion | Moosa Academy

A single number rarely tells the whole story. Forty students raised their hands — but out of how many? A ratio answers that by comparing two quantities through division, and it is the idea behind every percentage you have ever read.

Concept A ratio is a comparison by division

To form a ratio, divide the first quantity by the second.

\text{ratio} = \frac{\text{first number}}{\text{second number}}

So 3 to 4, 1 to 2 and 5 to 3 are all ratios. Division is the whole operation — nothing more complicated is hiding underneath.

Concept What the division means

Dividing means splitting something into equal parts. One apple shared between two people gives each person one half. A ratio does exactly this, whatever the numbers happen to be.

Concept Percentages are ratios out of 100

A percentage is simply a ratio whose second number is fixed at 100.

25\% = \frac{25}{100} \qquad 50\% = \frac{50}{100}

This is why percentages feel easy to compare: every one of them already shares the same denominator.

Concept Equivalent ratios

Ratios that reduce to the same value are called equivalent. These three look different but describe the same share:

\frac{1}{2} = 0.5

\frac{50}{100} = 0.5

\frac{200}{400} = 0.5

One apple between two people, 50%, and 200 out of 400 all come to one half. The numbers grow, but the relationship between them does not change.

Example Two test scores

A student answers 22 questions correctly out of 30. Another test has 100 questions and the student gets 85 right. What are the two scores as ratios?

First test:

22 \div 30 = 0.733\ldots

As a percentage, about 73.3%.

Second test:

85 \div 100 = 0.85

As a percentage, exactly 85%.

⟹ 73.3% and 85%

The second is easier to read off because the total is already 100 — the division does no real work. The first still needs dividing before it means anything.

Example A vote in the classroom

After a lesson, 40 students raise their hands to say they understood. Is that a good result?

On its own, 40 says nothing — the class total is missing.

Suppose the class has 60 students.

40 \div 60 = \frac{2}{3} \approx 0.667

As a percentage, about 66.7%.

⟹ about 66.7% understood

Had the class held 400 students, the same 40 hands would be only 10%. The count alone is never the answer; the total gives it meaning.

Note Mistakes to avoid

Reporting a count without the total it came from.

Dividing the numbers the wrong way round.

Assuming a bigger count always means a bigger share.

Treating a percentage as something other than a ratio out of 100.

Thinking equivalent ratios must use the same numbers.

Summary

A ratio compares two quantities by dividing one by the other.
Division splits a quantity into equal parts.
A percentage is a ratio whose second number is 100.
Equivalent ratios reduce to the same value: 1/2 = 50/100 = 200/400.
A count means little without the total it is measured against.