Summary: Comparing Quantities
Introduction
- In everyday life we often compare two quantities such as heights, weights, prices, speeds and so on.
- To compare two quantities meaningfully, the quantities must be of the same kind and expressed in the same units.
- There are several mathematical ways to compare quantities: ratios, proportions, and percentages. Each method is useful in different situations.
Ratios
Definition and Notation
- A ratio compares two quantities of the same kind by showing how many times one quantity is of the other.
- If two quantities are a and b (with b ≠ 0), their ratio can be written as a:b or as the fraction a/b (or using the division sign a ÷ b).
- A ratio can be simplified to its simplest form by dividing both terms by their greatest common divisor.
Equivalent Ratios and Simplest Form
- Two ratios are equivalent if they express the same relationship after simplification. For example, 2:3 is equivalent to 4:6 and to 6:9.
- To check equivalence, convert both ratios to fractions and compare them after simplification or use cross-multiplication.
Proportion
- Four numbers a, b, c, d are said to be in proportion if a:b = c:d. This is also written as a:b::c:d.
- If a:b = c:d, then ad = bc (this is the cross-multiplication rule).
Example - Checking a Proportion
Example: Are the four numbers 4, 9, 8, 18 in proportion?
Sol. Compare 4:9 and 8:18.
Compute cross-products.
4 × 18 = 72
9 × 8 = 72
Since the cross-products are equal, 4:9 = 8:18 and therefore the four numbers are in proportion.
Percentage - Another Way of Comparing Quantities
- Per cent comes from the Latin phrase per centum, meaning per hundred.
- A percentage is a fraction with denominator 100. The percentage sign % denotes "out of 100".
- For example, 1% = 1/100 = 0.01; 25% = 25/100 = 0.25; 150% = 150/100 = 1.5.
Converting Between Fractions, Decimals and Percentages
- To convert a fraction to a percent, multiply the fraction by 100 and add the % sign.
- To convert a percent to a decimal, drop the % sign and shift the decimal point two places to the left.
- To convert a decimal to a percent, shift the decimal point two places to the right and add the % sign.
- To convert a percent to a fraction, write the percent over 100 and simplify if possible.
Examples - Conversions
Example 1: Convert the fraction 3/8 into a percentage.
Sol. Multiply the fraction by 100.
(3/8) × 100 = 300/8
300/8 = 37.5
Therefore, 3/8 = 37.5%.
Example 2: Convert 12.5% into decimal and fraction.
Sol. Convert percent to decimal by shifting the decimal point two places to the left.
12.5% → 0.125
Convert percent to fraction by writing over 100 and simplifying.
12.5% = 12.5/100 = 125/1000 = 1/8.
Percentage Increase and Decrease
- Percentage increase tells how much a quantity has increased compared to the original amount, expressed as a percent.
- Percentage increase = (Amount of increase / Original amount) × 100.
- Percentage decrease tells how much a quantity has decreased compared to the original amount, expressed as a percent.
- Percentage decrease = (Amount of decrease / Original amount) × 100.
- When using these formulae, ensure the original amount is the correct base for comparison.
Example - Percentage Increase
Example: A book price rises from ₹120 to ₹150. Find the percentage increase.
Sol. Amount of increase = New price - Original price.
Amount of increase = 150 - 120 = 30.
Percentage increase = (Amount of increase / Original price) × 100.
Percentage increase = (30 / 120) × 100 = 25%.
Example - Percentage Decrease
Example: A dress is discounted from ₹2,000 to ₹1,600. Find the percentage decrease.
Sol. Amount of decrease = Original price - New price.
Amount of decrease = 2000 - 1600 = 400.
Percentage decrease = (Amount of decrease / Original price) × 100.
Percentage decrease = (400 / 2000) × 100 = 20%.
Buying and Selling (Profit and Loss)
- The cost price (CP) of an item is the price at which it is bought.
- The selling price (SP) of an item is the price at which it is sold.
- If SP > CP, there is a profit. Profit = SP - CP.
- If SP < CP, there is a loss. Loss = CP - SP.
- Profit percent = (Profit / CP) × 100.
- Loss percent = (Loss / CP) × 100.
- It is common and useful to remember that SP = CP + Profit and CP = SP - Loss.
Formulas and Useful Relations
- If profit percent = p%, then SP = CP × (1 + p/100).
- If loss percent = l%, then SP = CP × (1 - l/100).
Example - Profit Percent
Example: A trader buys a lamp for ₹800 and sells it for ₹1,000. Find the profit and profit percent.
Sol. Profit = SP - CP.
Profit = 1000 - 800 = 200.
Profit percent = (Profit / CP) × 100.
Profit percent = (200 / 800) × 100 = 25%.
Example - Loss Percent
Example: A shopkeeper buys an article for ₹450 and sells it at a loss of ₹90. Find the loss percent.
Sol. Loss = CP - SP.
Loss = 90.
Loss percent = (Loss / CP) × 100.
Loss percent = (90 / 450) × 100 = 20%.
Simple Interest
- The principal (P) is the sum of money borrowed or lent.
- The rate of interest (R) is usually given per cent per year (percent per annum).
- The time (T) is the period for which money is borrowed or lent, usually in years.
- The simple interest (SI) on a principal for a given rate and time is given by the formula SI = (P × R × T) / 100.
- The amount (A) to be paid after time T is A = P + SI.
Example - Simple Interest
Example: Find the simple interest on ₹5,000 at 6% per annum for 3 years. Also find the amount to be paid after 3 years.
Sol. Use the simple interest formula.
SI = (P × R × T) / 100.
SI = (5000 × 6 × 3) / 100.
Compute the product and divide by 100.
SI = (5000 × 18) / 100 = 90000 / 100 = 900.
Amount A = P + SI.
A = 5000 + 900 = 5900.
Therefore, the simple interest is ₹900 and the total amount is ₹5,900.
Converting Percentages to Fractions and Decimals (Revisited)
- To convert x% to a fraction, write x/100 and simplify if possible.
- To convert x% to a decimal, divide x by 100 or move the decimal point two places to the left.
- To convert a decimal to a percent, move the decimal point two places to the right and add the % sign.
Examples - Quick Conversions
Example: Express 0.375 as a percentage and as a fraction.
Sol.To get percentage, shift decimal point two places to the right.
0.375 → 37.5%.
To get fraction, write percentage over 100 and simplify, or convert decimal to fraction directly.
0.375 = 375/1000 = 3/8.
Example: Write 250% as a decimal and as a fraction.
Sol. As a decimal, move decimal point two places to the left.
250% → 2.50 → 2.5.
As a fraction, 250% = 250/100 = 5/2.
Summary and Tips for Students
- Always make sure quantities you compare are of the same kind and in the same units.
- Use ratios for direct comparisons, proportion to relate four quantities, and percentages when comparisons are to be expressed per hundred.
- Remember the principal formulae: SI = (P × R × T) / 100, Profit% = (Profit / CP) × 100, and Loss% = (Loss / CP) × 100.
- For percentage change problems, always divide by the original amount when computing percentage increase or decrease.
- Practice converting frequently between fractions, decimals and percentages to become quick and accurate.
FAQs on Summary: Comparing Quantities
| 1. What is the concept of comparing quantities in Class 7? |  |
| 2. How can we compare quantities using ratios and proportions in Class 7? | |
| 3. What are the key steps to solve problems related to comparing quantities in Class 7? | |
| 4. How can we compare quantities using percentages in Class 7? | |
| 5. Can we compare quantities using different units in Class 7? | |
