Ratios and Percentages: 24-Question Learning Review

Choose the model before calculating

This free, untimed review brings together percentages, ratios, proportional models, successive changes and weighted rates. Complete the four lessons first. Take as long as you need, and keep your working so that you can inspect the denominator, units and assumptions afterward.

There are 24 questions worth one mark each. A fully correct answer earns 1 mark; an incorrect or unanswered question earns 0. There is no pass/fail cutoff. Use your mistakes to decide what to revisit.

• Choose one: select exactly one option.
• Select all: select the complete correct set with no extra option. A partial set earns no credit.
• Enter a number: write the number in the requested unit. If a percentage number is requested, enter 15 for 15%, not 0.15. Follow any precision instruction in the question.

The questions mix methods deliberately. First identify the base or invariant, then calculate, and finally check whether the result makes sense.

Questions

Question 1 · Enter a number

What is 7.5% of 640? Enter the number.

Question 2 · Enter a number

₹720 is divided in the ratio 5:7. Enter the larger share in rupees.

Question 3 · Enter a number

800 rises by 10% and then falls by 10% of the raised value. Enter the final value.

Question 4 · Enter a number

In one group 40 people have a 75% success rate; in another 60 have a 50% rate. Enter the combined success percentage.

Question 5 · Choose one

36 is what percentage of 240?

A. 1.5%

B. 15%

C. 6.67%

D. 24%

Question 6 · Choose one

What is 1.5 km : 900 m in simplest form?

A. 2:3

B. 5:3

C. 3:2

D. 5:9

Question 7 · Choose one

After a 12% rise, a price is ₹896. What was it before the rise?

A. ₹784

B. ₹788.48

C. ₹800

D. ₹896

Question 8 · Choose one

Two nonempty groups have success rates 90% and 50%. Their sizes are not given. Which statement is justified?

A. The combined rate must be 70%

B. The combined rate must be 90%

C. The combined rate can exceed 90%

D. The exact combined rate is not determined

Question 9 · Enter a number

54 is 18% of a number. What is the number?

Question 10 · Enter a number

Positive values satisfy a:b=3:4 and b:c=2:5. If c=50, find a.

Question 11 · Choose one

A rate changes from 20% to 25%. Which pair is correct: percentage-point increase, relative percentage increase?

A. 5 points, 5%

B. 5 points, 25%

C. 25 points, 5%

D. 25 points, 25%

Question 12 · Enter a number

A batch of 500 items has 3% defective items; a batch of 1500 has 1%. Enter the combined defective percentage.

Question 13 · Choose one

A quantity rises from 25 to 30. What is its percentage increase?

A. 5%

B. 16.67%

C. 20%

D. 25%

Question 14 · Choose one

Six identical machines make 900 items in 5 hours. At the same constant independent rate, nine machines run for 4 hours. How many items do they make?

A. 900

B. 1080

C. 1200

D. 1350

Question 15 · Select all

X and Y are positive and X is 30% greater than Y. Select all correct statements.

A. X=1.3Y

B. Y=X/1.3

C. X−Y=0.3X

D. Y is (300/13)% less than X

Question 16 · Select all

Group A has 100 members with 40% success; B has 300 with 80% success. Select all true statements.

A. There are 280 successes in total

B. The combined success rate is 70%

C. The combined rate equals the unweighted mean 60%

D. Increasing B’s size while holding both group rates and A’s size fixed increases the combined rate

Question 17 · Select all

Select all values exactly equal to 3/8.

A. 37.5%

B. 0.375

C. 3.75%

D. 0.00375

Question 18 · Select all

Suppose y=kx for a fixed k>0 and x>0. Select all statements that must hold.

A. y/x is constant

B. Doubling x doubles y

C. xy is constant

D. The graph extends to a straight line through the origin

Question 19 · Enter a number

A quantity falls by 20% and then rises by 10% of the reduced value. What is its net percentage decrease? Enter the positive percentage number.

Question 20 · Choose one

Use the allocation pie chart and the within-group unused-rate table. How many tickets allocated to group B were used?

Chart data, accessible alternative: The pie chart represents 600 allocated tickets. Group A has 25%, B 45%, and C 30%. The accompanying table gives unused tickets as a percentage of each group’s own allocation: A 8%, B 10%, C 20%.

A. 27

B. 243

C. 270

D. 600

Question 21 · Choose one

Which interval contains 18% of 490?

A. 70 to 80, excluding 80

B. 80 to 90, excluding 90

C. 90 to 100, excluding 100

D. 100 to 110, excluding 110

Question 22 · Enter a number

At a constant speed of 75 km/h, how many minutes are needed for a 90 km journey?

Question 23 · Choose one

Unit price rises by 25%. To keep total spending fixed, by what percentage must quantity fall?

A. 20%

B. 25%

C. 30%

D. 33.33%

Question 24 · Choose one

Period 1 records 120 successes in 200 trials; period 2 records 162 in 300. What happened to the success rate?

A. It rose by 2 percentage points

B. It fell by 6 percentage points

C. It fell by 6% relative to the old rate

D. The rates cannot be compared

Answers and reasoning

Answer 1: 48

640×0.075=48. The 5% and 2.5% portions are 32 and 16.

Answer 2: 420

The total is 12 shares, each ₹60; 7 shares equal ₹420.

Answer 3: 792

800×1.10×0.90=792. The equal percentages do not cancel.

Answer 4: 60

Successes=30+30=60 out of 100, giving 60%.

Answer 5: B

36/240×100=15%. 1.5% is a decimal-place error.

• A: 1.5% would give 3.6, not 36, when applied to 240.
• B: 15% of 240 is exactly 36.
• C: 6.67% is approximately 16, not 36, of the base 240.
• D: 24% of 240 is 57.6, not 36.

Answer 6: B

1500:900=5:3 after converting both to metres.

• A: 1500/900 is greater than one, whereas 2/3 is less than one.
• B: 1500:900 simplifies to 5:3.
• C: 3:2=1.5, but 1500/900=5/3.
• D: 5:9 would result from an inconsistent conversion/simplification.

Answer 7: C

896/1.12=800. ₹788.48 is 896×0.88, an incorrect reversal.

• A: Subtracting 112 from the new price does not reverse a 12% multiplier.
• B: Multiplying 896 by 0.88 applies a new 12% decrease and does not undo the prior rise.
• C: 800×1.12=896.
• D: The unchanged value cannot produce 896 after a positive rise.

Answer 8: D

The combined rate depends on group-size weights and lies strictly between 50% and 90%. Equal weighting is not supplied.

• A: 70% needs equal weights; those are not given.
• B: A nonempty lower-rate group makes the combined rate strictly below 90%.
• C: A weighted average with positive weights cannot exceed the highest component rate.
• D: Different group sizes yield different combined rates, so no exact value follows.

Answer 9: 300

54/0.18=300; 0.18×300=54.

Answer 10: 15

b=50×2/5=20; a=20×3/4=15. Combined ratio is 3:4:10.

Answer 11: B

25−20=5 points; 5/20×100=25% relative increase.

• A: The point change is correct but 5% relative would move 20% to 21%, not 25%.
• B: The rate rises 5 points, which is 5/20=25% of the old rate.
• C: 25 is the relative percentage increase, not the point change.
• D: The point change is 5, not 25.

Answer 12: 1.5

There are 15+15=30 defective items out of 2000. 30/2000×100=1.5%.

Answer 13: C

Increase=5; original base=25. 5/25×100=20%. Dividing by 30 answers a different comparison.

• A: 5 is the absolute increase, not the percentage number.
• B: This divides 5 by the new value 30 rather than the old base 25.
• C: 5/25×100=20% is the requested increase.
• D: 25 is the old value, not the percentage increase.

Answer 14: B

Rate=900/(6×5)=30 items per machine-hour; 9×4×30=1080.

• A: This ignores the changed total productive machine-hours: 36 versus 30.
• B: 30 items per machine-hour ×36 machine-hours=1080.
• C: 1200 would require 40 machine-hours at the stated rate.
• D: This scales by machines alone and ignores that the run lasts 4 hours rather than 5.

Answer 15: A, B, D

X=1.3Y implies Y=X/1.3. The difference is 0.3Y, not 0.3X. Relative to X the gap is 0.3/1.3×100=300/13 percent.

• A: This is the direct translation of 30% greater than Y.
• B: Dividing X=1.3Y by 1.3 recovers Y.
• C: The gap is 0.3Y. If it were 0.3X, the implied comparison would differ.
• D: Relative gap=0.3Y/(1.3Y)×100=300/13 percent.

Answer 16: A, B, D

Successes=40+240=280 of 400, or 70%. Giving greater weight to the 80% group raises the weighted rate toward 80%.

• A: 100×0.4+300×0.8=280.
• B: 280/400×100=70%.
• C: The unweighted mean is 60%, but unequal group sizes make that inappropriate.
• D: More weight on the fixed 80% rate raises the aggregate toward 80%.

Answer 17: A, B

3/8=0.375=37.5%. In contrast, 3.75%=0.0375, and 0.00375 is smaller still.

• A: 37.5/100=0.375=3/8.
• B: 0.375=375/1000=3/8.
• C: 3.75%=0.0375, ten times smaller than 3/8.
• D: 0.00375 is one hundred times smaller than 3/8.

Answer 18: A, B, D

y/x=k; y(2x)=2kx; the linear relation has intercept zero. xy=kx² changes with x.

• A: For positive x, y/x=k.
• B: Replacing x by 2x gives y=2kx.
• C: xy=kx² varies with x; that is not the direct-proportion invariant.
• D: The algebraic line y=kx has zero intercept; the positive-domain ray lies on that extended line.

Answer 19: 12

0.8×1.1=0.88 of original, so the decrease is 12%.

Answer 20: B

The pie chart gives B 45% of 600 tickets, or 270. The mini-table gives 10% unused within B, so used=270×0.90=243.

• A: 27 is B’s unused count, not its used count.
• B: 45% of 600 is 270, and 90% of 270 is 243 used.
• C: 270 is B’s full allocation before unused tickets are removed.
• D: 600 is the total allocation to all groups.

Answer 21: B

0.18×490=88.2, so the second interval contains it.

• A: The exact result 88.2 exceeds this interval.
• B: 88.2 lies in [80,90).
• C: 88.2 is below 90.
• D: 88.2 is below 100.

Answer 22: 72

Time=90/75 h=1.2 h=72 min.

Answer 23: A

Quantity factor=1/1.25=0.8, a 20% fall.

• A: A factor 0.8 in quantity cancels a factor 1.25 in price.
• B: A 25% quantity fall gives 1.25×0.75=0.9375 of the original spending.
• C: A 30% fall makes spending 0.875 of original.
• D: This larger fall does not offset the price increase exactly.

Answer 24: B

Rates are 60% and 54%, so the fall is 6 percentage points, equivalent to a 10% relative fall. Absolute successes nevertheless rose by 42.

• A: The rate falls from 60% to 54%, so it does not rise.
• B: 54%−60%=−6 percentage points.
• C: The relative fall is 6/60=10%, not 6%.
• D: Counts and denominators for both periods are given, so their rates can be compared.

Use the result to choose your next step

• If you changed the denominator without noticing, revisit Percentages: Part, Whole and Comparison Base.
• If you combined ratios with different scales, mixed units, or assumed a rate model without evidence, revisit Ratios, Shares and Proportional Models.
• If you added successive percentage changes or reversed one by subtracting the same percentage, revisit Successive Changes and Reverse Percentages.
• If you averaged unequal-group rates or confused a total count with a rate, revisit Weighted Percentages and Data Cases.

For each missed question, write one line naming the decision that went wrong. Redo it without the solution, then check a different question using the same decision. A correct guess is still a reason to review the method.

Scope reference: GATE 2027 General Aptitude syllabus, page 1. This review focuses on the module's quantitative skills.

Analogy

Treat each problem like a labelled measuring job. The label tells you which quantity is the base, which relationship stays fixed, and which units belong to the result. A familiar number does not tell you which tool to use. A share split, a changing price and a combined success rate may all involve percentages but need different relationships. The analogy helps you pause and choose; it cannot supply a missing total, a productivity assumption or an unknown group size.

Quick reference

1. Base first: part = whole × percentage/100. A ratio needs compatible units and a scale to give absolute amounts.
2. Direct proportion keeps y/x fixed; inverse proportion keeps xy fixed, under the stated model.
3. Successive changes multiply. Reverse a nonzero multiplier by division.
4. Percentage-point change is new rate minus old rate; relative percentage change divides that difference by the old nonzero rate.
5. Combined rate = total numerator / total matching denominator. Equal weights guarantee a simple average for arbitrary rates; identical rates also remain unchanged under unequal weights.
6. Positive-weight averages lie between component extremes. Missing weights may prevent an exact answer.
7. A zero denominator blocks the division. State missing information or inconsistency rather than forcing a number.
8. Keep units, signs and precision until the final answer. Check the result against the model and a sensible bound.

Practice settings: free, untimed, 24 marks, +1 for a fully correct answer and 0 otherwise; no pass/fail cutoff.