Percentage and Its Applications
Percentage and Its Applications
Percentage appears in more questions than any other arithmetic idea, because profit-loss, interest and data interpretation all rest on it. Expect 2 direct questions and several indirect ones. The successive-change formula is the piece that decides the harder ones.
Conversions and the Fraction Table
Percentage = (Value ÷ Total) × 100
Converting percentages to fractions before calculating is the single biggest time saver in the arithmetic block.
| Percentage | Fraction | Percentage | Fraction |
|---|---|---|---|
| 50% | 1/2 | 12.5% | 1/8 |
| 25% | 1/4 | 16.67% | 1/6 |
| 75% | 3/4 | 20% | 1/5 |
| 33.33% | 1/3 | 6.25% | 1/16 |
| 66.67% | 2/3 | 37.5% | 3/8 |
Finding 37.5 per cent of 640 by decimal multiplication takes half a minute; as 3/8 of 640 it is 240, done in your head.
Percentage Change
% change = (Change ÷ Original value) × 100
The denominator is always the original value, never the new one. This is where most marks in the topic are lost.
✗ Price falls from 250 to 200, so the fall is 50/200 = 25% | ✓ The fall is 50/250 = 20% — divide by the original value
Note also that a rise and the corresponding fall are not the same percentage. Going from 200 to 250 is a 25 per cent rise, but coming back from 250 to 200 is a 20 per cent fall.
Successive Changes
Net % change = a + b + (a × b) ÷ 100
Use signs: a decrease is negative. A 20 per cent rise followed by a 20 per cent fall gives 20 − 20 − 400/100 = −4 per cent, a net loss of 4 per cent, not zero.
The general result is worth stating for teaching as well as for the exam: an equal rise and fall in either order always ends in a loss, and the loss is (x²/100) per cent.
Population, Depreciation and Error Percentages
Value after n periods = P (1 ± R/100)ⁿ
Plus for growth, minus for decline. This is the same formula as compound interest, which is why the two topics are really one.
- Population growth over years, and machine depreciation, both use it.
- If A is x per cent more than B, then B is less than A by 100x/(100 + x) per cent — not x per cent.
- If the price of an item rises by x per cent, consumption must fall by 100x/(100 + x) per cent to keep expenditure unchanged.
That asymmetric pair is asked directly and is one of the few formulas in the topic worth memorising rather than deriving.
प्रतिशत परिवर्तन हमेशा पुरानी कीमत से निकालें, नई से नहीं।
TRE pointer: Three items carry this topic in Part-III: the original-value denominator, the successive-change formula, and the 100x/(100+x) pair. The last one is the true discriminator, because candidates who have only done Part-II level percentage will answer x per cent and find that value waiting among the options. Given −1/3 negative marking, the fraction table is worth memorising purely for the time it frees up elsewhere.
60-Second Recap
- Convert percentages to fractions before calculating — 37.5% is 3/8.
- Percentage change always divides by the original value.
- A 25% rise reverses as a 20% fall — the two are not equal.
- Successive change: a + b + ab/100, with a decrease written negative.
- An equal rise and fall always leaves a net loss of x²/100 per cent.
- If A is x% more than B, B is 100x/(100+x) per cent less than A.
Frequently Asked Questions
Why is a twenty per cent rise followed by a twenty per cent fall not zero?
Because the fall applies to the increased amount, which is a larger base. A hundred rises to a hundred and twenty, then falls by twenty-four to ninety-six. The net effect is a four per cent loss, matching the formula x squared over a hundred.
If A is 25 per cent more than B, by how much is B less than A?
By twenty per cent, not twenty-five. Use the formula hundred x divided by a hundred plus x, which gives 2500 over 125, that is twenty. The intuitive answer of twenty-five per cent is always placed among the options to catch a quick reading.
Which denominator do I use for percentage change?
Always the original value, the one you started from. If a price moves from 250 to 200, the change of fifty is divided by 250, giving twenty per cent, not by 200. Using the new value is the most common single error in the whole topic.
Is the population growth formula the same as compound interest?
Yes, in structure. Value after n periods equals the starting value times one plus or minus the rate over a hundred, raised to the power n. Growth takes the plus sign and depreciation the minus. Recognising this saves learning two separate formulas.