Comparing Quantities Class 7 Maths Notes

Key Points to Remember

• A ratio is a comparison of two quantities of the same kind by division. It must always be expressed in its simplest form.
• To compare two ratios, convert them into equivalent fractions with the same denominator.
• The unitary method involves finding the value of a single unit first, then calculating the value for the required number of units.
• Percentage means "per hundred" or "out of hundred". It is denoted by the symbol '%'.
• To convert a fraction or decimal to a percentage, multiply by 100%.
• To convert a percentage to a fraction or decimal, divide by 100.
• Profit occurs when Selling Price (SP) > Cost Price (CP). Profit = SP - CP.
• Loss occurs when Cost Price (CP) > Selling Price (SP). Loss = CP - SP.
• Profit % = (Profit / CP) × 100 and Loss % = (Loss / CP) × 100.
• Simple Interest (SI) = (P × R × T) / 100, where P = Principal, R = Rate (per annum), T = Time (in years).

Essential Definitions

Ratio

A comparison of two quantities of the same unit and kind by division. It is written as a:b or a/b.

Equivalent Ratios

Ratios that have the same value, even if they look different. For example, 1:2 and 2:4 are equivalent ratios.

Unitary Method

A technique where you first find the value of a single unit (e.g., cost of one item) and then use it to find the value of the required number of units.

Percentage

A way of expressing a number as a fraction of 100. It is denoted by the symbol '%'. For example, 25% means 25 out of 100.

Cost Price (CP)

The price at which an article is purchased.

Selling Price (SP)

The price at which an article is sold.

Profit

The amount gained when the selling price is greater than the cost price (SP > CP).

Loss

The amount incurred when the cost price is greater than the selling price (CP > SP).

Principal (P)

The original sum of money deposited or borrowed.

Interest (I)

The extra money paid for using borrowed money, or the money earned on invested money.

Amount (A)

The total money returned at the end of the period, which includes the Principal and the Interest (A = P + I).

Ratios: The Foundation of Comparison

Ratios are a fundamental concept in comparing quantities. A ratio is essentially a mathematical way to compare two quantities of the same kind by division. For instance, if you have 3 apples and 5 oranges, the ratio of apples to oranges is 3:5. It's crucial that the quantities being compared are of the same units; if not, you must convert them before forming a ratio. For example, to compare 50 cm to 2 meters, you first convert 2 meters to 200 cm, making the ratio 50 cm : 200 cm, which simplifies to 1:4.

Always express ratios in their simplest form. This means dividing both parts of the ratio by their highest common factor (HCF). For example, the ratio 15:20 simplifies to 3:4 by dividing both by 5. Ratios do not have units because the units cancel out during division. Equivalent ratios are ratios that represent the same comparison, even if the numbers are different. For instance, 1:2, 2:4, and 5:10 are all equivalent ratios. You can find equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero number. Understanding ratios is key, as they form the basis for understanding proportions and percentages.

Ratio vs. Percentage: A Quick Comparison

Steps to Solve Percentage Problems

1. Understand the Question — Identify what is given (e.g., percentage, total quantity, part of a quantity) and what needs to be found (e.g., percentage of a number, the whole quantity, percentage increase/decrease).
2. Formulate the Equation/Relation — Remember that 'x% of Y' means (x/100) Y. For increase/decrease, it's (Original Value ± Change) / Original Value 100.
3. Perform Calculations — Substitute the known values into your equation and solve for the unknown. Be careful with decimal and fraction conversions.
4. Check Your Answer — Does the answer make sense in the context of the problem? For instance, if you found a 10% increase, the new value should be greater than the original.

Worked Examples

• {"title":"Example 1: Ratio Simplification","bodyMarkdown":"Q: Simplify the ratio 75 paise to ₹4.\nA: First, convert to the same unit. ₹4 = 400 paise.\nRatio = 75 paise : 400 paise\nDivide both by their HCF, which is 25.\n75 ÷ 25 : 400 ÷ 25 = 3 : 16."}
• {"title":"Example 2: Finding Percentage of a Quantity","bodyMarkdown":"Q: Find 12% of 600 kg.\nA: 12% of 600 = (12/100) × 600\n= 12 × 6 = 72 kg."}
• {"title":"Example 3: Simple Interest Calculation","bodyMarkdown":"Q: Calculate the simple interest on a principal of ₹8000 at an interest rate of 5% per annum for 3 years.\nA: Given: P = ₹8000, R = 5%, T = 3 years.\nSI = (P × R × T) / 100\nSI = (8000 × 5 × 3) / 100\nSI = (80 × 5 × 3) = 400 × 3 = ₹1200."}

Exam Tip: Word Problems & Unit Conversion

For word problems in 'Comparing Quantities', always read carefully to identify what quantities are being compared, what is the whole, and what is the part. A common mistake is failing to convert units before forming a ratio or calculating percentages. Ensure all quantities are in the same unit (e.g., both in cm, both in rupees, both in hours) before proceeding. Also, remember that profit/loss percentages are always calculated on the Cost Price (CP) unless specified otherwise. Clearly state the formulas you use in your solutions to score full marks.

Practice Questions with Solutions

• Q: What is the ratio of 1 metre to 25 centimetres in its simplest form? A: 1 metre = 100 centimetres. So, 100:25 = 4:1.
• Q: Convert 3/5 into a percentage. A: (3/5) × 100% = 60%.
• Q: If a shopkeeper bought a toy for ₹150 and sold it for ₹180, what is the profit percentage? A: Profit = SP - CP = 180 - 150 = ₹30. Profit % = (30/150) × 100 = 20%.
• Q: Find the Amount if Principal = ₹2000, Rate = 6% p.a., Time = 2 years. A: SI = (2000 × 6 × 2) / 100 = ₹240. Amount = P + SI = 2000 + 240 = ₹2240.

Frequently Asked Questions