Percentages and Successive Changes

What you will learn

A percentage compares an amount with a named base, taking that base as 100%. The calculation becomes reliable when you can say what the 100% represents. In this lesson you will convert exact representations, find a part or a missing whole, calculate and reverse a change, follow successive changes, and distinguish a percentage-point difference from a relative percentage change.

You need Fractions Decimals and Exact Calculation and Ratio and Proportion. Direct and inverse proportion comes earlier in our recommended sequence, but inverse proportion is not required for this lesson. We use invented quantities and rates for learning, not live prices, population claims or financial advice. Interest and compounding periods belong to the next module.

1 Retrieve fractions and ratio parts

Try before reading the feedback.

1. Write 0.4 as a simplest fraction.
2. Find 3/4 of 80.
3. A:B = 2:3. What fraction of A+B is A?

Feedback: 0.4 = 4/10 = 2/5. Three quarters of 80 is 3 × (80÷4) = 60. A is 2 of the 2+3 = 5 equal parts, so its fraction of the whole is 2/5. Using 2/3 in the third answer would compare A with B, not A with the total.

2 Percent is a fraction with a named base

The symbol % means per hundred. Thus p% = p/100. If the base amount is B, then p% of B = (p/100) × B. The number p is the numerical percentage, not the multiplier: 8% uses multiplier 0.08, not 8.

Equivalent fractions, decimals and percentages represent the same share. Multiply a fraction or decimal by 100 to obtain its numerical percentage. Divide the numerical percentage by 100 to obtain the fraction or decimal.

A percentage does not specify an amount on its own. For example, 10% of 50 is 5, while 10% of 200 is 20. The rate is the same; the bases differ. Percentages can exceed 100 when comparing a larger amount with a smaller base: 150% of 20 is 30. However, a genuine subset of a positive whole cannot exceed that whole, so its share is at most 100%.

Worked example 1 Convert without losing exactness

Convert 3/8 and 0.075 to percentages; express 12.5% as a fraction.

3/8 × 100 = 300/8 = 37.5, so 3/8 = 37.5%. 0.075 × 100 = 7.5, so 0.075 = 7.5%. 12.5% = 12.5/100 = 125/1000 = 1/8.

Check: 37.5/100 = 0.375 = 3/8; 7.5/100 = 0.075. Writing 0.075 as 75% changes its value by a factor of 10. Likewise, 12.5% is 0.125, not 1.25.

For a non-terminating result, keep an exact fraction unless rounding is requested: 1/3 = 33 1/3%. The decimal 33.33% is only an approximation. A useful mental method is to combine familiar shares of the same base: 15% of B = 10% of B + 5% of B. This addition is valid because both percentages refer to B.

3 Part, percentage and whole are different unknowns

Start from one relationship: Part = (percentage/100) × base.

• To find the part, multiply the given base by the percentage fraction
• To find the percentage, divide part by the positive base, then multiply by 100
• To find the base from a positive percentage, divide the part by that percentage fraction

The “part” in this equation is the amount being compared; it can exceed the base in a comparison above 100%. In literal whole-sharing questions, use the actual total as base.

Worked example 2 Ratio to percentage of the whole

A basket has only red and blue balls in the ratio 3:7, with 200 balls in total. Find the red count and the percentage that are red.

Total ratio parts = 3+7 = 10. The red fraction of the whole is 3/10. Red count = (3/10) × 200 = 60. Red percentage = (3/10) × 100 = 30%.

Check: 60/200 × 100 = 30%. Dividing 3 by 7 would express red relative to blue, a different comparison. The blue share is 70%, and the two exhaustive shares add to 100%.

Worked example 3 Find the rate, then find a missing whole

A 45 g amount belongs to a 180 g whole. What percentage is it? Separately, 36 is 15% of what number?

First calculation: percentage = (45/180) × 100 = (1/4) × 100 = 25%. Both amounts use grams, so the units cancel.

For the second, let the unknown whole be B. Then 36 = (15/100) × B. Divide by 15/100: B = 36 × 100/15 = 240. Alternatively, 15% corresponds to 36, so 1% corresponds to 36/15 = 2.4 and 100% corresponds to 240.

Check: 15% of 240 = 0.15 × 240 = 36. Taking 15% of 36 would give 5.4, but the question asks for the original whole. If a percentage is zero, this reverse division is not available: zero is 0% of many possible bases.

Checkpoint A

1. Find 12.5% of 88.
2. 45 is 18% of what number?
3. A:B = 2:3. Express A as a percentage of A+B.

Feedback: 12.5%=1/8, so 88÷8=11. The missing whole is 45÷0.18=250; check 250×0.18=45. In the last question A is 2/5 of the whole, so its percentage is 40%, not (2/3)×100%. If you chose the wrong denominator, label the whole before repeating the calculation.

4 A change is measured against the original amount

For a positive original amount B and final amount F: Percentage change = ((F−B)/B) × 100.

A positive result is an increase; a negative result is a decrease. You can instead use the positive decrease B−F and explicitly say “decrease”. The denominator is the starting base B, not the final value F.

A rise of r% adds (r/100)B to B: New amount = B + (r/100)B = B(1+r/100). A fall of d% removes (d/100)B: New amount = B − (d/100)B = B(1−d/100).

These factors explain the method. For example, a 15% rise makes the new amount 115% of the old, so its factor is 1.15. It does not make the new amount merely 15% of the old.

Worked example 4 Forward rate and reverse change

A quantity changes from 160 to 184. Find its percentage increase. A different quantity is 414 after a 15% rise. Find that quantity's original value.

The first increase is 184−160 = 24. Percentage increase = (24/160) × 100 = 15%. The old 160 is the reference.

For the reverse problem, the final 414 is 115% of the original. Therefore original = 414÷1.15 = 360. Check: 360 + 15% of 360 = 360+54 = 414.

Subtracting 15% of 414 would use the wrong base: it gives 414−62.1 = 351.9. Reversal undoes a multiplication by division; it does not simply reverse the sign of the percentage.

For ordinary nonnegative quantities, a decrease can range from 0% to 100%. A 100% decrease gives zero. You cannot recover a unique original from that zero by dividing by the zero factor. Also, percentage change from a zero original is undefined because its denominator would be zero. Report the actual change instead; do not invent an infinite percentage.

5 Successive changes use successive bases

Write a base ledger: original → after first change → after second change. Each percentage acts on the amount specified in its own stage. When it acts on the latest amount, use the latest amount as its base.

For an original B, a rise r% followed by a fall d% gives: Final = B(1+r/100)(1−d/100). To get the net percentage relative to the original, subtract 1 from the combined factor and multiply by 100. This factor method follows the two actual calculations; do not memorise an unlabelled “add the percentages” trick.

Worked example 5 An increase, then a decrease

Start at 500. Increase by 20%, then decrease the new value by 10%.

First base is 500: increase = 0.20×500 = 100, so the next value is 600. Second base is 600: decrease = 0.10×600 = 60, so final value is 540. Net increase = 540−500 = 40, which is (40/500)×100 = 8%.

The combined factor is 1.20×0.90 = 1.08. Check: 500×1.08=540. Subtracting 10 from 20 would predict 10%, but that incorrectly treats both changes as percentages of 500. Same-base percentage amounts may be added; successive percentages on different bases need their bases tracked.

Worked example 6 Equal opposite rates and restoration

Start at 320, rise 25%, then fall 25% of the new value.

320×1.25 = 400, then 400×0.75 = 300. The rise was 80, but the fall was 100. Thus net decrease = 20, or (20/320)×100 = 6.25%. Check the combined factor: 1.25×0.75 = 0.9375, so 93.75% of the original remains.

Now consider a separate restoration problem: after a 25% fall, what percentage rise restores the original? Use an original of 100 as a convenient scale. The reduced value is 75; the needed increase is 25. Relative to the new base 75, the required rate is (25/75)×100 = 33 1/3%.

The restoration factor is 100/75 = 4/3, and 0.75×(4/3)=1. A 25% rise would only give 75×1.25=93.75, so it does not restore the original. The choice of 100 makes the comparison easy; the conclusion applies to every positive original under these same percentage changes.

6 Reverse comparisons change the denominator

“A is r% more than B” uses B as base. Asking how much less B is than A uses A as base. The numerical difference may be the same, but the denominator is not.

Worked example 7 More than and less than are not symmetric

A is 25% larger than B. By what percent is B smaller than A?

Choose B=100 units, so A=125 units. The difference is 25 units. For “B smaller than A”, divide by A: (25/125)×100 = 20%.

Check: 80% of 125 = 100. Claiming 25% in both directions would give 75% of 125 = 93.75, not 100. The chosen units are a convenient scaling of the relationship, not extra data about the actual quantities.

7 Percentage points compare rates; relative percentages compare change

When two values already are percentages, subtracting their numerical values gives a difference in percentage points. For a relative percentage change, divide that difference by the original numerical percentage and multiply by 100, provided the original is not zero.

Worked example 8 The same change with two different descriptions

A stated rate changes from 40% to 50%. Find the percentage-point increase and the relative percentage increase.

Percentage-point increase = 50−40 = 10 percentage points. Relative increase = ((50−40)/40)×100 = 25%.

Both descriptions are correct but describe different comparisons. On a common 100-unit reference, the amounts rise from 40 to 50; the extra 10 is one quarter of the old 40. Do not call a 10-point increase a 10% relative increase.

Checkpoint B

1. A value is 270 after a 10% fall. Find the original.
2. A value rises 20%, then falls 5% of the new value. Find the net change.
3. Can a percentage change be calculated in the usual way for an amount rising from 0 to 7?

Feedback: (1) 270÷0.9=300; check 300×0.9=270. (2) 1.2×0.95=1.14, so the net increase is 14%. (3) No: the actual increase is 7, but division by the zero starting base is undefined. Keep these domain checks alongside the arithmetic.

8 Independent practice

Attempt the twelve questions before reading the key. State the 100% base next to each calculation. These are original untimed learning questions: +1 for correct, 0 otherwise, with no negative marking or pass cutoff. They are not a full RRB CBT simulation.

1. Express 0.035 as a percentage.

A. 0.35% B. 3.5% C. 35% D. 350%

2. Express 18.75% as an exact simplest fraction.

A. 3/8 B. 15/8 C. 3/16 D. 75/16

3. What is 35% of 240?

A. 84 B. 35 C. 156 D. 8400

4. 54 is what percentage of 216?

A. 400% B. 54% C. 75% D. 25%

5. 63 is 14% of which number?

A. 8.82 B. 450 C. 882 D. 49

6. For positive A and B, A:B=4:11. What exact percentage of A+B is A?

A. 36 4/11% B. 73 1/3% C. 26 2/3% D. 4%

7. A value falls from 250 to 210. What is the percentage decrease?

A. 16% B. 19 1/21% C. 40% D. 84%

8. A positive value is 432 after a 20% rise. What was the original value?

A. 345.6 B. 518.4 C. 412 D. 360

9. A positive quantity rises 10%, then rises 20% of its new value. What is the net percentage increase?

A. 30% B. 32% C. 2% D. 132%

10. A positive quantity falls 20%, then rises 20% of its reduced value. What is the net result relative to the original?

A. No change B. 4% increase C. 4% decrease D. 40% decrease

11. A is 40% more than positive B. By what exact percentage is B less than A?

A. 28 4/7% B. 40% C. 71 3/7% D. 60%

12. A rate changes from 30% to 36%. Which pair correctly gives the percentage-point increase and relative percentage increase, in that order?

A. 6 percentage points; 6% B. 20 percentage points; 6% C. 6 percentage points; 120% D. 6 percentage points; 20%

9 Explained answer key

1. B. Multiply by 100 to get the numerical percentage: 0.035×100=3.5, so 0.035=3.5%. Check 3.5÷100=0.035. A multiplies by only 10; C by 1,000; D by 10,000. The percent sign already supplies division by 100 when converting back.

2. C. 18.75%=18.75/100=1875/10000=3/16 after division by 625. Check 3÷16=0.1875. A equals 37.5%, twice the required share. B equals 1.875=187.5%, ten times the required share. D divides 18.75 by 4 instead of 100; 75/16 is greater than 1 and cannot represent this percentage below 100%.

3. A. 35% of 240=(35/100)×240=84. Mentally, 30% is 72 and 5% is 12, giving 84 because both use the same base. Check 84/240=0.35. B repeats the percentage number. C is the remaining 65% of 240. D forgets division by 100.

4. D. 216 is the base. Percentage=(54/216)×100=(1/4)×100=25%. Check 25% of 216=54. A reverses the fraction, finding 216 as a percentage of 54. B confuses the amount with its percentage. C is the complementary 162/216 share, not the requested 54/216.

5. B. Let the base be B. Then 0.14B=63, so B=63/0.14=450. Check 450×0.14=63. A takes 14% of 63, reversing the task. C multiplies 63 by 14 without percentage scaling. D subtracts 14 as a bare amount. A fraction below 1 of the base is 63, so the base must exceed 63.

6. C. The whole has 4+11=15 ratio parts. A is 4/15 of it, so its percentage is (4/15)×100=80/3%=26 2/3%. A uses B alone as the denominator: 400/11%=36 4/11%. B is the percentage of the whole belonging to B. D treats four ratio parts as four out of a hundred.

7. A. Decrease=250−210=40. Divide by the original 250: 40/250×100=16%. Check 250×0.84=210. B uses final 210 as base, giving 400/21%. C calls the amount 40 a percentage. D is the percentage remaining, 210/250×100, not the decrease.

8. D. The final is 120% of the original, so original=432÷1.2=360. Check 360+72=432. A subtracts 20% of the final, using the wrong base. B increases the final again by 20%. C subtracts the bare number 20. To undo multiplying by 1.2, divide by 1.2.

9. B. Use factors 1.10×1.20=1.32. The final is 132% of the original, so the increase is 32%. With base 100, the sequence is 100→110→132. A adds rates despite the changed base. C reports only the extra 2 percentage points above the naive 30% sum, not the complete 32% increase. D reports the final percentage, not the increase.

10. C. The combined factor is 0.80×1.20=0.96, so 96% remains and the decrease is 4%. For original 100, the values are 100→80→96. A wrongly cancels equal opposite rates. B gets the size but reverses the direction. D adds the two rate numbers without their signs or bases.

11. A. Choose B=100 and A=140. For B less than A, the base is A, so percentage=(40/140)×100=200/7%=28 4/7%. Check B/A=5/7, so the shortfall is 2/7. B keeps the original comparison base. C gives B as a percentage of A, not the shortfall. D subtracts 40 from 100 without changing base.

12. D. Points=36−30=6. Relative increase=(6/30)×100=20%. Check 30×1.20=36. A confuses the point difference with relative change. B swaps unlike descriptions and still mislabels 6. C gives the final rate as 120% of the old one, instead of the 20% increase.

10 Sources and scope

RRB CEN 09/2025 §14.1, printed p.28, lists Percentages. This lesson develops that topic; it does not predict exam frequency. Concept references: NCERT Ganita Prakash, Grade 8 Part II, Chapter 1, §§1.1–1.3, printed pp.1–16 and the successive-percentage example on p.26; NIOS Secondary Mathematics 211, Chapter 8, §§8.1–8.4, printed pp.204–212. Hindi terminology cross-check: NIOS Hindi Chapter 8, §§8.1–8.4, printed pp.212–221. Percentage-point comparison is our brief transfer explanation, not a separately named RRB topic or a copied textbook exercise. All teaching, numerical cases and questions are original. No interest formula, current tax rate or source scan is included.

Analogy

Imagine a measuring strip divided into 100 equal shares and label its entire length “the base”. If that base length is 240 cm, each 1% share is 2.4 cm; if the base is 500 cm, each 1% share is 5 cm. The strip explains why the same percentage can represent different amounts. After a change, a percentage of the new length needs a newly labelled 100% strip.

Limit: this is a representation, not a claim that every object can be physically cut into 100 pieces. A comparison can extend beyond the strip, such as 150% of the base. Percentage points compare the numerical markings of two rates; they do not by themselves specify the underlying amounts.

Quick reference

• p% = p/100; name the amount that represents 100%
• Part=(p/100)×base
• p=(part/base)×100 for positive base
• Base=part÷(p/100) when p>0
• For A:B=a:b, A as a percentage of A+B is a/(a+b)×100
• Relative change=(final−original)/original×100, original>0
• Rise r%: factor 1+r/100; fall d%: factor 1−d/100
• Reverse a change by dividing by its nonzero factor
• Successive changes: multiply the stated stage factors, then compare with 1
• Equal opposite percentage changes generally do not cancel
• In reverse more/less comparisons, change the denominator to the named reference
• Percentage points=new rate−old rate; relative rate change=(new−old)/old×100, old≠0
• Zero original: usual percentage change undefined; 100% fall: original not uniquely recoverable
• Keep repeating results exact or label approximation; same-base parts may be added