Ratio questions: methods and worked examples

What a ratio tells you

A ratio of 3:2 between two teams means that for every 3 people in the first team there are 2 in the second. The total is 3 + 2 = 5 parts.

From those parts you can read three different things, and tests check that you keep them apart:

• Each group as a share of the whole: the first team is 3/5 of the total (60%) and the second is 2/5 (40%).
• One group compared with the other: the first team is 3/2 = 1.5 times the size of the second.
• The difference: the first team is 1 part bigger than the second.

The trap: reading 3:2 as "the first group is 3/2 of the total" or "the first group is 3/5 of the second". Before you calculate, say in words which comparison the question asks for: part to part, or part to whole.

Order matters. "The ratio of managers to staff is 1:6" puts managers first. If you swap the order, every answer that follows is wrong, and the swapped answer is usually one of the options.

Sharing an amount in a ratio

This is the most common ratio question in job tests. Use three steps every time:

1. Add the parts.
2. Divide the total by the parts to find one part.
3. Multiply by each group's parts.

Worked example: a $4,800 bonus pool is split between two teams in the ratio 5:3. How much does each team get?

1. 5 + 3 = 8 parts.
2. $4,800 / 8 = $600 per part.
3. First team: 5 x $600 = $3,000. Second team: 3 x $600 = $1,800.

Check: $3,000 + $1,800 = $4,800. Always add your shares back up; it takes two seconds and catches most slips.

Three-way shares work the same way. 360 hours of project time are split across design, build and testing in the ratio 2:3:7. That is 12 parts, so one part is 360 / 12 = 30 hours. Design gets 60, build 90 and testing 210. The check: 60 + 90 + 210 = 360.

When you know one share, not the total

Many ratio problems give you one group's size, or the difference between two groups, and ask for something else. The "value of one part" step still works; you just find the part from a different number.

From one group: the ratio of supervisors to operators on a site is 2:9, and there are 54 operators. How many supervisors?

• 9 parts = 54, so 1 part = 6.
• Supervisors = 2 x 6 = 12. The whole site has 11 parts = 66 people.

From the difference: two partners share profits 7:4, and the first receives $45,000 more than the second. How much do they share in total?

• The difference is 7 - 4 = 3 parts, so 3 parts = $45,000 and 1 part = $15,000.
• The total is 11 parts = $165,000. The shares are $105,000 and $60,000; their difference is $45,000, which confirms the answer.

The trap: dividing a known group by the total parts. In the site example, 54 / 11 is not a part; 54 is the operators' 9 parts alone.

Simplifying and scaling ratios

Simplify by dividing every number by the largest number that goes into all of them. 45:60 divides by 15 to give 3:4. 84:126 divides by 42 to give 2:3. If you cannot see the largest factor, divide by small ones in turn: 84:126, halve to 42:63, divide by 3 to 14:21, divide by 7 to 2:3.

Decimals and fractions in a ratio: multiply every term by the same number until they are whole. 1.5:2 becomes 3:4 (times 2). 1/2:1/3 becomes 3:2 (times 6).

Scaling up or down: a cleaning solution is mixed 1:4, concentrate to water.

• For a set total: 2.5 liters of solution is 5 parts, so 1 part is 0.5 liters: 0.5 liters of concentrate and 2 liters of water.
• From one ingredient: 3 liters of concentrate needs 3 x 4 = 12 liters of water, making 15 liters in all.

The trap in scaling: adding the same amount to both sides. Going from 1:4 to 2:5 by "adding one to each" changes the mix. Ratios scale by multiplying, never by adding.

Combining ratios and changing ratios

Combining two ratios that share a term. You are told that A:B = 2:3 and B:C = 4:5, and asked for A:B:C. The two B values (3 and 4) do not match, so scale both ratios until B is the same.

1. Find a common value for B. The lowest number both 3 and 4 divide into is 12.
2. Scale A:B by 4 to get 8:12, and B:C by 3 to get 12:15.
3. Read across: A:B:C = 8:12:15, so A:C = 8:15.

The trap answer joins the ratios without matching B, giving 2:3:5 or 2:4:5.

Changing ratios. A box holds 40 items, red and blue, in the ratio 3:5. That is 15 red and 25 blue. If 5 red items are added, the new ratio is 20:25, which simplifies to 4:5. The method is always the same: turn the ratio into real numbers first, apply the change to the numbers, then turn it back into a ratio. You cannot add items to a ratio directly.

Working backward: how many red items must be added to make the ratio 1:1? Blue stays at 25, so red must reach 25: add 10.

Ratios, fractions and percentages

Numerical reasoning questions often switch between the three forms in one question. The conversions:

• Ratio to fraction of the whole: 3:5 means the first group is 3/(3 + 5) = 3/8 of the total.
• Fraction to percentage: 3/8 = 0.375 = 37.5%.
• Percentage to ratio: if 35% of applicants are internal, the rest (65%) are external, so internal to external is 35:65 = 7:13.

The trap: treating a ratio's numbers as a fraction directly. In a team where women to men is 3:5, women are 37.5% of the team, not 3/5 = 60%. Test writers include 60% because it is the answer you get if you skip the "add the parts" step.

For the percentage side of these conversions, including percentage change and reverse percentages, see our percentage questions guide.

Proportion and rates from tables

Direct proportion: when one quantity doubles, the other doubles. 4 printers produce 600 pages in 10 minutes, so 1 printer produces 150 pages and 6 printers produce 6 x 150 = 900 pages in the same time. Find the rate for one, then multiply.

Inverse proportion: when more of one quantity means less of the other. 6 workers finish a job in 8 days, so the job is 6 x 8 = 48 worker-days. With 4 workers it takes 48 / 4 = 12 days. The trap is scaling the wrong way: fewer workers cannot mean fewer days.

Rates from tables. Many numerical reasoning questions hide a ratio in a data table. For example, a table lists three teams:

• Team A: 12 staff, 2 managers
• Team B: 20 staff, 4 managers
• Team C: 21 staff, 3 managers

"Which team has the most staff per manager?" Turn each into a unit rate: A is 6:1, B is 5:1 and C is 7:1, so Team C. Comparing the raw staff numbers would point you to Team B, which is the trap.

Speed is also a rate, so the same thinking covers distance over time. Our speed, distance and time guide covers that topic in full.

A fast method for ratio questions under time pressure

1. Label the parts. Write the ratio with its names, such as "S 2 : O 9", so the order cannot flip.
2. Mark what you know: a total, one group, or a difference. That decides what you divide by: all the parts, one group's parts, or the difference in parts.
3. Find one part, then multiply. Nearly every ratio question is these two moves.
4. Use the options. In a 5:3 split, the larger share must be a multiple of 5 parts and the total must divide by 8. Options that break this can be crossed out before you calculate.
5. Check by adding back. Shares should add to the total; a difference in parts should match the stated difference.

Ratios often appear together with percentages and averages in the same data set. If you are preparing for a timed numerical test, practice moving between the three until it feels routine.