COMPARING QUANTITIES
COMPARING QUANTITIES: RATIOS, PERCENTAGES & MORE
Introduction
We compare quantities in our daily life all the time—whether it’s comparing prices while shopping, calculating discounts, understanding profit and loss, or checking interest rates in banks. This chapter will help you understand different ways to compare numbers, use ratios and percentages effectively, and apply these concepts in real-life situations.
1. Ratios and Proportions
A ratio is a way to compare two quantities by division. It helps express how much of one quantity exists compared to another.
Example:
If a class has 20 boys and 10 girls, the ratio of boys to girls is:
Ratio=20:10=2:1
This means for every 2 boys, there is 1 girl.
Smart Trick to Remember:
- Always write a ratio in the simplest form.
- Order matters! (e.g., 2:5 is different from 5:2).
- Multiply or divide both terms of a ratio by the same number to find equivalent ratios.
Proportion
A proportion states that two ratios are equal.
Example:
If 2 pens cost ₹10, how much will 6 pens cost?
We set up the proportion:
10/2=x/6
Cross multiply:
2x=60
x=₹30
So, 6 pens will cost ₹30.
2. Percentages – A Way of Comparing
A percentage means per hundred. It helps in making fair comparisons between different quantities.
Formula for Percentage
Percentage=(WholePart)×100
Example:
A student scores 45 marks out of 50 in an exam. What is the percentage?
45/50×100=90%
So, the student scored 90%.
Smart Trick to Remember:
- To convert a fraction into a percentage, multiply by 100.
- To convert a percentage into a fraction, divide by 100.
- 50% = 1/2, 25% = 1/4, 75% = 3/4 (Memorize these for quick calculations).
3. Increase and Decrease Percentages
When prices go up or down, we use percentage increase or decrease.
Formula:
Percentage Change=(Change in Value/ Original Value)×100
Example:
A shirt’s price was ₹500 but is now ₹600. Find the percentage increase.
{(600−500)/500}×100={100/500}×100=20%
So, the price increased by 20%.
Smart Trick to Remember:
- If the new value is more than the original, it’s a percentage increase.
- If the new value is less, it’s a percentage decrease.
4. Discounts – Saving Money!
Shops often give discounts during sales. A discount is a percentage of the original price.
Formula for Discount:
{Discount} ={Discount %}/{100} ×{Marked Price}
Example:
A ₹1000 mobile is available at a 20% discount. What is the final price?
20/100×1000=200
So, the final price = ₹1000 – ₹200 = ₹800.
Smart Trick to Remember:
- A higher percentage discount means more savings.
- Always calculate a discount on the Marked Price (M.P.).
5. Profit and Loss – Business Calculations
Every business earns or loses money. Profit and Loss calculations are important for business owners.
Formulas:
Profit=Selling Price (SP)−Cost Price (CP)
Loss=Cost Price (CP)−Selling Price (SP)
{Profit %} ={Profit}/{CP}×100
{Loss %} = {Loss}/{CP}×100
Example:
A shopkeeper buys a bag for ₹500 and sells it for ₹650. Find the profit percentage.
Profit=650−500=150
{Profit %} ={150}/{500}× 100 = 30%
So, the profit is 30%.
Smart Trick to Remember:
- SP > CP → Profit, CP > SP → Loss.
- More profit means a better business!
6. Simple Interest – Earning More Money!
Banks give interest on money deposited. This is called Simple Interest (SI).
Formula:
SI=P×R×T/100
Where,
- P = Principal (Initial Amount)
- R = Rate of Interest
- T = Time in years
Example:
If ₹5000 is deposited in a bank at 5% interest per year for 3 years, find the interest earned.
SI=(P×R×T)/100
SI=(5000×5×3)/100=750
So, the bank gives ₹750 as interest.
Smart Trick to Remember:
- More time = More interest earned.
- Banks use Compound Interest for better savings.
Summary – Key Takeaways
✔ Ratios help compare quantities.
✔ Proportions show two equal ratios.
✔ Percentages are useful in exams, shopping, and business.
✔ Profit and Loss help businesses track money.
✔ Simple Interest is a way to earn extra money in banks.
Mind Map for Quick Revision
📌 Ratios & Proportions → Compare numbers
📌 Percentages → Express as per 100
📌 Profit/Loss → Business calculations
📌 Discounts → Save money on shopping
📌 Simple Interest → Earn extra money in banks
Conclusion
Comparing quantities is a skill we use every day. Whether it’s understanding discounts, bank interest, or profits in business, these concepts help in real life. Mastering these topics will make calculations easier and smarter. Keep practicing and applying these in daily life to become a math genius!
📝 Practice MCQs (50 Questions)
Ratios & Proportions (1–10)
- Ratio of 10:5 in simplest form:
A) 5:2
B) 2:1
C) 1:2
D) 10:5 - Ratio of 20:30:
A) 2:3
B) 3:2
C) 1:2
D) 4:5 - Equivalent ratio of 2:3:
A) 4:6
B) 3:4
C) 5:6
D) 6:5 - Ratio 5:1 means:
A) Equal
B) 5 times
C) Half
D) Double - If 3:4 = x:8, x =
A) 6
B) 8
C) 4
D) 5 - 2:5 :: 4:x → x =
A) 10
B) 8
C) 6
D) 5 - Ratio of boys to girls = 2:1, total 30 → boys =
A) 10
B) 15
C) 20
D) 25 - Ratio order matters:
A) True
B) False
C) Maybe
D) None - 6:3 simplifies to:
A) 3:1
B) 2:1
C) 1:2
D) 6:3 - Proportion means:
A) Inequality
B) Equality of ratios
C) Sum
D) Difference
Percentages (11–20)
- Percentage means:
A) Per 10
B) Per 100
C) Per 1000
D) None - 50% =
A) 1/2
B) 1/3
C) 2/3
D) 3/4 - 25% =
A) 1/2
B) 1/4
C) 3/4
D) 2/3 - 75% =
A) 1/2
B) 3/4
C) 1/4
D) 2/3 - 40 out of 50 → % =
A) 70%
B) 80%
C) 90%
D) 60% - 60% of 200 =
A) 100
B) 120
C) 140
D) 160 - 10% of 500 =
A) 50
B) 100
C) 40
D) 60 - 150% means:
A) 1.5
B) 0.5
C) 2
D) 3 - Convert 0.25 to %:
A) 25%
B) 50%
C) 75%
D) 20% - 200% equals:
A) 1
B) 2
C) 3
D) 4
Increase/Decrease & Discount (21–30)
- Price increases from 100 to 120 → % increase =
A) 10%
B) 20%
C) 30%
D) 40% - Price decreases from 200 to 150 → % decrease =
A) 20%
B) 25%
C) 30%
D) 15% - Discount is calculated on:
A) CP
B) SP
C) MP
D) None - 10% discount on ₹1000 =
A) 100
B) 200
C) 50
D) 150 - Final price after ₹200 discount on ₹1000:
A) 700
B) 800
C) 900
D) 600 - Higher discount means:
A) Less saving
B) More saving
C) Equal
D) None - New price > old price →
A) Decrease
B) Increase
C) Equal
D) None - New price < old price →
A) Increase
B) Decrease
C) Equal
D) None - 25% of ₹400 =
A) 100
B) 200
C) 150
D) 50 - 50% discount means:
A) Half price
B) Double
C) Same
D) None
Profit & Loss (31–40)
- Profit =
A) CP − SP
B) SP − CP
C) SP + CP
D) None - Loss =
A) SP − CP
B) CP − SP
C) Sum
D) None - CP = 500, SP = 600 → Profit =
A) 100
B) 200
C) 50
D) 150 - CP = 700, SP = 600 → Loss =
A) 100
B) 200
C) 50
D) 150 - Profit % formula:
A) Profit/SP
B) Profit/CP ×100
C) CP/SP
D) None - CP = 500, Profit = 100 → % =
A) 10%
B) 20%
C) 25%
D) 30% - SP > CP means:
A) Loss
B) Profit
C) Equal
D) None - CP > SP means:
A) Profit
B) Loss
C) Equal
D) None - Selling price = CP →
A) Profit
B) Loss
C) No profit no loss
D) None - Profit increases when:
A) SP increases
B) CP increases
C) Both decrease
D) None
Simple Interest (41–50)
- SI formula includes:
A) P, R, T
B) Only P
C) Only R
D) Only T - Principal means:
A) Interest
B) Amount invested
C) Rate
D) Time - Rate is:
A) Time
B) Percentage
C) Money
D) None - Time is measured in:
A) Months
B) Years
C) Days
D) Seconds - SI on ₹1000 at 10% for 1 year:
A) 100
B) 200
C) 150
D) 50 - SI increases with:
A) Time
B) Rate
C) Principal
D) All - SI on ₹2000, 5%, 2 years:
A) 200
B) 300
C) 400
D) 100 - More time →
A) Less interest
B) More interest
C) Same
D) None - Banks give:
A) Loss
B) Interest
C) Profit
D) None - SI is:
A) Complex
B) Simple
C) Double
D) None
✅ Answer Key
1-B, 2-A, 3-A, 4-B, 5-A
6-A, 7-C, 8-A, 9-B, 10-B
11-B, 12-A, 13-B, 14-B, 15-B
16-B, 17-A, 18-A, 19-A, 20-B
21-B, 22-B, 23-C, 24-A, 25-B
26-B, 27-B, 28-B, 29-A, 30-A
31-B, 32-B, 33-A, 34-A, 35-B
36-B, 37-B, 38-B, 39-C, 40-A
41-A, 42-B, 43-B, 44-B, 45-A
46-D, 47-A, 48-B, 49-B, 50-B
🎯 Smart Tips for Revision
- 💡 Ratio → Compare numbers
- 💡 % → Out of 100
- 💡 Profit → SP > CP
- 💡 Loss → CP > SP
- 💡 SI → P × R × T
