Ratio Formula — How to Find, Simplify & Divide

What Is the Ratio Formula?

A ratio measures how many times one quantity contains another, or how a whole is split between parts. Because $a : b = \frac{a}{b}$, every ratio is also a fraction, and most ratio work is fraction work in disguise.

$$\boxed{a : b = \frac{a}{b}\quad(b \neq 0)}$$

Three things follow directly from the formula, and they cover most questions readers ask:

• Simplifying a ratio. Divide both terms by their greatest common factor (GCF). $12 : 18 = \frac{12}{18} = \frac{2}{3} = 2 : 3$.

• Equivalent ratios. Multiply or divide both terms by the same non-zero number. $2 : 3 = 4 : 6 = 6 : 9$ — all the same comparison, like equivalent fractions.

• Dividing a quantity in a ratio. To split a total in the ratio $a : b$, add the parts ($a + b$), divide the total by that, then give each share its number of parts. This is the part of the topic most worksheets test, so it gets its own examples below.

Both quantities in a ratio must share the same unit before you compare them — you cannot put $2$ metres against $3$ centimetres without converting first. Ratio sits right next to proportion, which is a statement that two ratios are equal.

How Do You Divide an Amount in a Given Ratio?

This is the question that shows up most in exams: "Divide $$200$ between two people in the ratio $3 : 5$." The method is three clean steps.

1. Add the parts. $3 + 5 = 8$ total parts.

2. Find one part. $$200 \div 8 = $25$ per part.

3. Multiply out each share. $3 \times$25 = $75$ and $5 \times$25 = $125$.

The check is that the shares add back to the total: $$75 + $125 = $200$. That add-back is the safety net for the whole topic — if the shares do not return the original amount, something went wrong.

Examples of the Ratio Formula

Example 1

A class has 12 boys and 18 girls. Write the ratio of boys to girls in simplest form.

$$12 : 18 = \frac{12}{18} = \frac{2}{3} = 2 : 3.$$

Final answer: $2 : 3$.

Example 2

Divide 200 in the ratio 3 : 5.

Wrong attempt. A student divides $200$ by $3$ and by $5$ to get $66.7$ and $40$, then stops. Check it: $66.7 + 40 = 106.7$, nowhere near $200$. Dividing by each term separately ignores that the parts must share the whole.

Correct. Add the parts first: $3 + 5 = 8$. One part is $200 \div 8 = 25$. The shares are $3 \times 25 = 75$ and $5 \times 25 = 125$.

Final answer: $75$ and $125$, which add to $200$. ✓

Example 3

Is $4 : 6$ equivalent to $6 : 9$?

Simplify both: $4 : 6 = \frac{2}{3}$ and $6 : 9 = \frac{2}{3}$. Equal fractions mean equal ratios.

Final answer: Yes, both equal $2 : 3$.

Example 4

The ratio of two numbers is $5 : 7$ and their sum is $96$. Find the numbers.

Total parts $= 5 + 7 = 12$, so one part $= 96 \div 12 = 8$. The numbers are $5 \times 8 = 40$ and $7 \times 8 = 56$.

Final answer: $40$ and $56$.

Example 5

A 3 : 2 ratio of flour to sugar uses 600 g of flour. How much sugar?

Set the ratios equal (a proportion): $\frac{3}{2} = \frac{600}{x}$. Cross-multiply: $3x = 1200$, so $x = 400$. The same splitting logic powers any cost-price calculation where a marked amount is shared across cost and profit.

Final answer: $400$ g of sugar.

Example 6

Split a $$1{,}500$ profit between three partners in the ratio $2 : 3 : 5$.

The three-term ratio works the same way. Total parts $= 2 + 3 + 5 = 10$; one part $=$1{,}500 \div 10 = $150$. Shares: $2 \times 150 =$300$,$3 \times 150 = $450$, $5 \times 150 =$750$.

Final answer: 300$, 450$,$$750$ — adding to $$1{,}500$. ✓

Why Ratios Matter — From Kitchens to Bank Balance Sheets

Ratios were formalised because comparison by division is how scaling works, and scaling is everywhere.

• Maps and scale models. A map scale of $1 : 50{,}000$ means one unit on paper is $50{,}000$ in the world — pure ratio, no addition.

• Cooking and chemistry. A recipe or a chemical reaction holds because the proportion between ingredients is fixed; double both terms and it still works, add to one and it breaks.

• Finance. Companies are read through ratios — the debt-to-equity ratio, the current ratio, the price-to-earnings ratio — each a single number standing in for a whole comparison.

• The golden ratio. The proportion $1 : 1.618\ldots$ recurs in art, architecture, and the spiral of a sunflower seed head — the most famous ratio in mathematics.

The bigger idea waiting downstream is rates — speed (distance to time), density (mass to volume), exchange rates (one currency to another). A rate is just a ratio between quantities of different units, which is why ratio is the foundation under so much of later science.

Where Ratios Go Sideways

Mistake 1: Treating a ratio as a total

Where it slips in: "Divide in a ratio" problems.

Don't do this: Divide the total by each term of the ratio separately and call those the shares.

The correct way: Add the parts to find the total number of parts, find the value of one part, then multiply.

Mistake 2: Comparing quantities in different units

Where it slips in: Ratios of two lengths, weights, or amounts given in mixed units.

Don't do this: Write the ratio of $2$ m to $50$ cm as $2 : 50$.

The correct way: Convert to a common unit first. $2$ m $= 200$ cm, so the ratio is $200 : 50 = 4 : 1$.

Mistake 3: Reversing the order of the ratio

Where it slips in: "Ratio of A to B" versus "ratio of B to A."

Don't do this: Write the ratio of girls to boys when the question asked for boys to girls — order matters in a ratio.

Conclusion

• The ratio formula writes a comparison of two like quantities as $a : b = \frac{a}{b}$, with $b \neq 0$.

• Simplify a ratio by dividing both terms by their GCF; build equivalent ratios by multiplying both terms by the same number.

• To divide an amount in a ratio, add the parts, find one part, then multiply each share — and check the shares add back to the total.

• The most common mistake is treating the ratio as a total and dividing by each term separately.

• Ratio is the foundation under rates, proportion, scale, and the financial ratios used to read a balance sheet.

Try It Yourself — Three Problems

1. Simplify the ratio $24 : 36$.

2. Divide $$450$ between two people in the ratio $4 : 5$.

3. The ratio of two ages is $3 : 7$ and their sum is $50$. Find both ages.

Answer to Question 1: $2 : 3$. Solve Questions 2 and 3 with the add-the-parts method; if the shares do not add back to the total, return to Mistake 1.

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