GMAT Ratios

What a ratio means

A ratio compares quantities multiplicatively. The ratio 3:5 means that for every three units of the first quantity, there are five of the second. It can represent 3 and 5, 6 and 10, or 30 and 50. The ratio alone does not specify the actual size of either quantity.

Order matters. If the ratio of red to blue items is 3:5, there are three red items for every five blue items. Reversing it to 5:3 changes the relationship. Labels prevent this error: write red:blue = 3:5, then map red to 3k and blue to 5k.

Distinguish part-to-part from part-to-whole. If red:blue is 3:5, the whole has 8 total parts. Red is 3/8 of the total and blue is 5/8. The ratio red:total is 3:8, not 3:5. Many ratio errors come from answering a fraction-of-total question with the original part-to-part ratio.

Use a common multiplier

Represent ratio parts with the same multiplier k. If apples:oranges = 4:7, write apples = 4k and oranges = 7k. If there are 55 fruits total, then 11k = 55, so k = 5, with 20 apples and 35 oranges. Check that the quantities preserve 4:7 and add to 55.

If the difference is known, subtract the ratio parts. Suppose the ratio of adults to children at an event is 7:4 and there are 21 more adults than children. The difference is 3k, so 3k = 21 and k = 7. There are 49 adults and 28 children. Using the total parts, 11k, would be wrong because the prompt gives a difference.

If one quantity is known, use its part to find k. If the number of adults is 42 in a 7:4 ratio, then 7k = 42 and k = 6, so children number 24. Avoid assuming the smaller ratio term is one actual person; it is one ratio unit, which can represent several people.

Combine ratios with a shared term

To combine ratios, make the shared quantity equal in both relationships. If A:B = 2:3 and B:C = 4:5, the B parts are 3 and 4. Scale the first ratio by 4 and the second by 3: A:B = 8:12 and B:C = 12:15. Thus A:B:C = 8:12:15.

A common mistake is to write 2:3:5 because the numerals appear in order. That ignores the different sizes represented by B. Always align the shared term before joining ratios. If the ratios use different units, convert or establish a relationship before combining them.

Original worked problems

Part-to-whole percentage

A department has engineers and analysts in a ratio of 5:3. What fraction of the department are engineers? There are 8 total ratio parts, so engineers are 5/8 of the department, or 62.5 percent. The answer is not 5/3, which is the engineer-to-analyst ratio, nor 5/8 expressed as 58 percent.

A useful check is to verify that the fraction is between zero and one because it describes a portion of the whole. The denominator includes both groups. If the question asked how many engineers per analyst, 5/3 would be appropriate.

A ratio changes after adding an amount

The ratio of red to blue marbles is 2:5. After 6 red marbles are added, the ratio becomes 1:2. How many marbles were there originally? Let red = 2k and blue = 5k. Then (2k + 6)/(5k) = 1/2. Cross-multiply: 4k + 12 = 5k, so k = 12. Originally there were 7k = 84 marbles.

A frequent error adds 6 to both colors. The prompt adds red marbles only, so only the red quantity changes. Another error sets 2k/5k = 1/2, overlooking that the ratio after the change is different from the original one.

Combine two ratios

If A:B = 3:4 and B:C = 6:7, what is A:C? Scale the first ratio so B equals 12: A:B = 9:12. Scale the second so B equals 12: B:C = 12:14. Therefore A:C = 9:14. No actual value for B is needed because the common term is aligned.

A distractor may multiply the outer terms and answer 18:28. That ratio simplifies to 9:14, so it happens to be equivalent, but the direct alignment method is easier to explain and less prone to mistakes when more than two ratios are involved.

Ratios, rates and units

A rate is a ratio with different units, such as 60 miles per hour. Keep the unit attached to each quantity. If a solution has 12 grams of ingredient per 3 liters, simplify to 4 grams per liter. If you compare 90 miles per hour with 2 miles per minute, convert one unit before comparing: 2 miles per minute equals 120 miles per hour.

For a mixture, a ratio of concentrate to water of 1:4 means one part concentrate and four parts water, five total parts. Concentrate is one fifth of the mixture. If the 1:4 ratio instead describes concentrate to finished solution, then it is one quarter of the finished solution. Read precisely what each side names.

When quantities are scaled together, ratios stay constant. If a recipe for 4 people requires 3 cups of rice, a recipe for 10 people requires 3 x 10/4 = 7.5 cups, assuming portions scale proportionally. If a fixed amount is added to only one ingredient, the ratio changes; represent the original amounts with k before applying the change.

Proportions and cross multiplication

A proportion states that two ratios are equal, such as a/b = c/d. When denominators are nonzero, cross multiplication gives ad = bc. Use it to solve unknown values, but preserve the meaning of the variables and units. A proportion can be algebraically correct and still model the wrong relationship if the quantities are paired incorrectly.

Suppose 8 workers complete 120 units in 5 hours at a constant rate. How many units do 10 workers complete in 6 hours? Output is proportional to workers and time: 120 x (10/8) x (6/5) = 180 units. A one-step proportion would be insufficient if it omitted either the worker count or the duration.

Direct proportion means both quantities rise or fall together at a fixed ratio. Inverse proportion means their product stays constant, as when more identical workers take less time to finish a fixed job. Identify which relationship the problem describes before cross multiplying.

Common ratio traps

• Reversing the order of the quantities.
• Using part-to-part terms as if they were part-to-whole fractions.
• Combining ratios without matching their shared term.
• Applying a change to every quantity when the prompt changes only one.
• Treating a ratio as fixed actual counts instead of scalable parts.
• Comparing rates without converting units.
• Assuming direct proportion when a fixed amount or inverse relationship applies.

Draw a small labeled table when several groups or stages appear. If a ratio changes, use separate variables for the before and after relationships, or express both in the same k where appropriate. Verify the final counts are whole numbers when the story requires people or objects. A fractional answer may indicate a setup error or an impossible condition.

Practice without a calculator

GMAT Quant has no calculator. Ratios often simplify mental arithmetic: reduce 18:30 to 3:5, or recognize that 5/8 is 62.5 percent. Keep fractions until the last step. Multiplying decimals early can make arithmetic harder and obscure the structure.

Practice by translating varied word problems rather than memorizing a single phrase. Make a ratio table for mixtures, use k for counts, align shared terms for chained ratios, and write units for rates. After each answer, substitute the quantities into the original relationship and confirm it holds.

When reviewing a miss, identify whether the issue was order, total parts, scaling, unit conversion or a changing ratio. Then solve a new problem with a different setting. The skill transfers when you can see that a recipe, staffing plan and investment allocation all use the same proportional reasoning.

A ratio after a change

When a problem changes one part, keep the original multiplier and alter only the quantity named. Suppose the ratio of adults to children is 3:2. If 6 children leave, the new ratio becomes 3:1. Let adults = 3k and children = 2k. Then 3k/(2k - 6) = 3. Solving gives 3k = 6k - 18, so k = 6. The original group had 18 adults and 12 children.

Check the new counts: after 6 children leave, there are 18 adults and 6 children, a 3:1 ratio. This substitution catches sign errors. If the prompt instead added adults, only the adult expression would change. Write 'before' and 'after' beside the ratios when a story has multiple stages.

If both quantities change, make each change explicit. If a group gains 4 adults and 2 children, the new ratio is (3k + 4):(2k + 2). Do not assume the common multiplier remains a whole number unless the problem requires integer counts; solve the equation, then check that the resulting counts make sense.