Percentage, Ratio and Proportion
Percentage, Ratio and Proportion
Percentage and ratio together are the backbone of the arithmetic block, and they reappear inside profit-loss, interest and data interpretation. Expect 1 to 2 direct questions plus two or three more that quietly depend on these ideas. Learn the conversions well and the rest of the section speeds up.
Percentage Basics and Conversions
Percent means per hundred. To convert, multiply a fraction by 100 for a percentage, and divide a percentage by 100 for a fraction.
Percentage = (Value ÷ Total) × 100
If 34 students out of 850 in a Bihar middle school are absent, the absence rate is (34 ÷ 850) × 100 = 4%.
Percentage Increase and Decrease
% change = (Change ÷ Original value) × 100
The word original is the whole test. The denominator is always the starting value, never the new one.
✗ Price rises 200 → 250, so rise = 50/250 = 20% | ✓ Price rises 200 → 250, so rise = 50/200 = 25%
For two changes one after the other, do not add them. Use the successive formula.
Net % = a + b + (a × b) ÷ 100
A 10% rise followed by a 10% fall gives 10 − 10 + (10 × −10) ÷ 100 = −1%, that is a 1% net loss — not zero. This sign trap is asked again and again.
Ratio and Proportion
A ratio compares two quantities of the same kind. To divide an amount in a ratio, add the parts and share out.
Divide Rs 4,500 between two teachers in the ratio 4 : 5. Total parts = 9, so one part = 500. The shares are Rs 2,000 and Rs 2,500.
| Concept | Rule | Example |
|---|---|---|
| Equivalent ratio | Multiply or divide both terms | 2:3 = 4:6 |
| Proportion | a:b = c:d means ad = bc | 2:3 = 6:9 |
| Mean proportional | √(a × b) | Between 4 and 9 → 6 |
| Third proportional | b² ÷ a | Of 4 and 6 → 9 |
| Compound ratio | Multiply term by term | 2:3 and 4:5 → 8:15 |
प्रतिशत परिवर्तन में हर बार पुरानी कीमत से भाग दें, नई से नहीं।
TRE pointer: In a teaching exam the percentage question is often wrapped in a school context — enrolment change, attendance, pass percentage or scheme coverage in a district. The examiner's favourite trap is the successive-change question where candidates add the two percentages and pick the wrong option. Also note that with five options and −1/3 for every wrong answer, a guessed percentage question costs more than it used to in the four-option era.
60-Second Recap
- Percentage = (value ÷ total) × 100; percent change always divides by the original value.
- Successive change: a + b + ab/100. A 10% rise then 10% fall is a 1% net loss, not zero.
- To divide in a ratio, add the parts and find the value of one part.
- Mean proportional between a and b is √(ab); third proportional is b²/a.
- Fraction-percent equivalents from the previous topic cut solving time in half.
- Expect school-context wrappers: attendance, enrolment, pass percentage.
Frequently Asked Questions
What is the successive percentage change formula?
Net percentage change equals a + b + (a × b) ÷ 100, where a and b are the two changes with their signs. A decrease is written as a negative number. This formula prevents the most common error in the topic, which is simply adding the two percentage figures together.
Why is a 10% increase followed by a 10% decrease not zero?
Because the second change is applied to a bigger base. A 10% rise on 100 gives 110, and a 10% fall on 110 removes 11, not 10, leaving 99. The net effect is a 1% loss. The formula a + b + ab/100 captures this shift of base automatically.
How do I find the mean proportional between two numbers?
The mean proportional between a and b is the square root of their product, √(a × b). Between 4 and 9 it is √36 = 6. Do not confuse it with the third proportional, which is b² ÷ a and gives 9 when the numbers are 4 and 6.
Is ratio asked separately or mixed with other topics?
Both. You will usually get one direct ratio question, but ratio also sits inside profit-loss, average, partnership and data interpretation questions. Treating it as a tool rather than a chapter is the practical approach for the Part-II General Studies maths block.