Chapter 8 — Comparing Quantities

Meaning: Per cent (or percent) means "per hundred". The symbol used is %. If we say 45%, it means 45 out of every 100 or 45 per 100.

In other words, a percent is a special fraction with denominator 100. So 45% = 45/100. We can also write a percent as a decimal by dividing by 100: 45% = 0.45.

How to read and use percent:

• Percent to fraction: write the number over 100 and simplify. Example: 20% = 20/100 = 1/5.
• Percent to decimal: divide by 100. Example: 12% = 0.12.
• To find what percent a part is of a whole: percent = (part ÷ whole) × 100.
• To find the part when percent of whole is known: part = (percent ÷ 100) × whole.

Common percent equivalents: 50% = 1/2 = 0.5, 25% = 1/4 = 0.25, 75% = 3/4 = 0.75, 10% = 1/10 = 0.1, 100% = 1.

Why percent is useful: Percent is used to compare quantities easily when totals differ, for example comparing marks, prices, discounts, interest rates, population proportions, and survey results. Converting to percent makes comparisons intuitive because all values are expressed per 100.

Overview: Fractions, decimals and percentages are three ways to represent parts of a whole. Converting among them helps compare quantities easily in everyday life (shopping, marks, recipes, statistics).

Basic ideas:

• Fraction a/b means a parts out of b equal parts of a whole.
• Decimal is another way of writing parts using base 10 (place values: tenths, hundredths, thousandths…).
• Percent means per hundred ("out of 100"). 1% = 1/100.

How to convert:

• Fraction → Decimal: Divide numerator by denominator (a ÷ b). If the division stops, the decimal is terminating; if it repeats, the decimal is recurring. Example: 3/4 = 3 ÷ 4 = 0.75.
• Decimal → Fraction: Write the decimal as a fraction using place value and simplify. Example: 0.125 = 125/1000 = 1/8.
• Fraction → Percent: Convert fraction to decimal (a ÷ b), then multiply by 100 and add % sign; or directly multiply fraction by 100%. Example: 3/4 = 0.75 × 100% = 75%.
• Percent → Fraction: Write percent over 100 and simplify: 45% = 45/100 = 9/20. Alternatively, treat percent as "percent of 100".
• Decimal → Percent: Multiply decimal by 100 and add % sign. Example: 0.45 × 100 = 45%.
• Percent → Decimal: Divide percent value by 100 or move decimal point two places left. Example: 12.5% = 12.5 ÷ 100 = 0.125.

Notes on simplification and repeating decimals: After converting to a fraction, always simplify (divide numerator and denominator by GCD). For recurring decimals (for example 0.666...), you can write 0.666... = 2/3; converting repeating decimals to fractions uses algebraic methods taught in higher classes.

Quick conversion tips:

• To turn a fraction with denominator 2,4,5,8,10,20,25,50,100 easily into a decimal/percent — memorize common equivalents (e.g., 1/4 = 0.25 = 25%).
• To convert percent to decimal, move decimal point two places left; to convert decimal to percent, move two places right.

Meaning: Expressing one quantity as a percentage of another means finding how many parts per 100 the first quantity (part) is of the second quantity (whole). We write this as (part ÷ whole) × 100%.

Steps to find percentage:

1. Identify the part and the whole.
2. Divide the part by the whole to get a fraction or decimal.
3. Multiply the result by 100 to convert it into a percentage and add the % sign.

Notes:

• If the part is less than the whole, the percentage is less than 100%.
• If the part equals the whole, the percentage is 100%.
• If the part is greater than the whole, the percentage is more than 100%.

Conversions you should know: to convert a fraction or decimal to percent multiply by 100; to convert percent to decimal divide by 100.

Practical uses: marks in exams, discounts and taxes, population comparisons, ingredient proportions in recipes, and comparing heights or weights.

What is percentage increase and decrease?

Percentage increase or decrease measures how much a quantity grows or shrinks compared to its original value, expressed as a percentage. We always compare the change to the original (or base) value.

Key ideas:

• Original (base) value: the starting amount.
• Change: the amount added (increase) or removed (decrease).
• Percent change: (change ÷ original) × 100%.

Interpretation: A positive percent change means an increase; a negative percent change means a decrease.

Quick ways to get the new value: If the original value is P and the percent change is r%:

• After an increase of r%: New = P × (1 + r/100).
• After a decrease of r%: New = P × (1 − r/100).

Reverse problem: If you know the new value and the percent change, you can find the original: Original = New ÷ (1 ± r/100) (use + for increase, − for decrease).

Important note for successive changes: Percent increases and decreases are not additive. If a quantity increases by a% and then decreases by b%, the final value is P × (1 + a/100) × (1 − b/100). Two equal opposite percent changes do not return you to the original value (for example, +10% then −10% results in a net decrease).

Successive Percentage Changes occur when a quantity is increased or decreased by a percentage more than once, one after another. You do not add or subtract the percentages directly. Instead, convert each percentage change into a multiplier (factor) and multiply the factors to get the final value.

Rules: Increase by p% → multiplier = (1 + p/100). Decrease by p% → multiplier = (1 - p/100). For two successive changes p% and q%: final value = original × (1 + p/100) × (1 + q/100). The net percentage change = [(final / original) − 1] × 100%.

Important points: (1) Successive changes are multiplicative, not additive, so equal increase and decrease by the same percentage do not cancel (e.g., 10% up then 10% down results in a 1% fall). (2) The order of successive changes does not affect the final result because multiplication is commutative.

What the topic means: Sometimes we are given a quantity after it has changed by some percentage (it may have increased or decreased) and we need to find the original quantity before the change. This uses the idea of a percentage multiplier.

Key idea (multiplier): If an original quantity is O and it increases by p%, the new (final) quantity F = O × (1 + p/100). If it decreases by p%, F = O × (1 - p/100). To find O when F and p are known, divide F by the multiplier.

1. When there is a percentage increase: O = F ÷ (1 + p/100). Example: after a 20% increase the final value is 600. Original = 600 ÷ 1.20 = 500.
2. When there is a percentage decrease: O = F ÷ (1 - p/100). Example: after a 15% decrease the final value is 255. Original = 255 ÷ 0.85 = 300.

Steps to solve:

1. Identify whether the change is an increase or a decrease and note p%.
2. Compute the multiplier: 1 + p/100 for increase, 1 - p/100 for decrease.
3. Divide the given final quantity by the multiplier to get the original quantity.
4. Check by applying the percentage to your answer to see if you get the final quantity.

Important notes: For a percentage decrease, p must be less than 100 (because a 100% decrease makes the original zero). For successive percentage changes, divide by each multiplier in reverse order.

Profit and Loss is a basic money-related topic in the Class 7 chapter Comparing Quantities. When an item is bought at a Cost Price (CP) and sold at a Selling Price (SP), the difference between SP and CP tells us whether there is a profit or a loss.

Definitions:

• Profit (or gain) occurs when SP > CP. Profit = SP − CP.
• Loss occurs when SP < CP. Loss = CP − SP.
• Profit percent and Loss percent measure profit or loss as a percentage of the cost price:

• Profit% = (Profit / CP) × 100
• Loss% = (Loss / CP) × 100

Use CP (the base) when computing percent. Rearranging the formulas helps to find SP or CP when percent is given:

• If profit% is p, SP = CP × (1 + p/100).
• If loss% is l, SP = CP × (1 − l/100).

Important points for Class 7 students:

• Always compare SP and CP first to decide profit or loss.
• Percent is always taken with respect to CP (not SP) in this chapter.
• Round answers sensibly when dealing with currency (usually to two decimal places for rupees and paise).

These ideas are applied in many everyday situations such as shops, marketplaces, garage sales and discounts. Understanding how profit and loss relate to percentages helps compare deals and make choices.

Marked Price (MP) is the price written on an article by the shopkeeper (also called the list price or labeled price). Selling Price (SP) is the actual price at which the article is sold to the customer.

Discount is the reduction given on the marked price. It can be expressed as an amount (in rupees) or as a percentage of the marked price. In simple terms:

• Discount (amount) = Marked Price − Selling Price
• Discount percentage = (Discount ÷ Marked Price) × 100%

To find the selling price after a discount, either subtract the discount amount from the marked price or calculate directly using the discount rate. For example, a 20% discount means the buyer pays 80% of the marked price.

Key points for students:

• If MP and discount% are given, SP = MP × (1 − discount%/100).
• If SP and discount% are given and you need MP, MP = SP ÷ (1 − discount%/100).
• Discount% is always taken on the marked price, not on the selling price.

What is a percentage? A percentage is a way to express a part of a whole as parts per 100. It helps compare quantities easily in real life (discounts, taxes, marks, population, etc.).

Basic ideas and conversions

• Fraction to percentage: multiply by 100. Example: 3/4 = (3/4)×100 = 75%.
• Decimal to percentage: multiply by 100. Example: 0.65 = 65%.
• Percentage to fraction: divide by 100 and simplify. Example: 20% = 20/100 = 1/5.

Finding part, whole or percentage

• To find what percent a part is of a whole: percentage = (part ÷ whole) × 100.
• To find the part from given percentage: part = (percentage ÷ 100) × whole.
• To find the whole when part and percentage are known: whole = (part × 100) ÷ percentage.

Percentage increase and decrease

• Percentage change = ((new − old) ÷ old) × 100. If the result is positive it is an increase; if negative it is a decrease.
• To calculate a new value after p% increase: new = old × (1 + p/100). For p% decrease: new = old × (1 − p/100).
• For successive changes, multiply the factors. Example: increase by p% then decrease by q% gives factor (1 + p/100)×(1 − q/100).

Everyday contexts where percentages are used

• Shopping: discounts and sale prices (percentage off), and adding sales tax/GST (percentage on price).
• Finance: simple interest or savings rates expressed as percent per year (basic idea).
• Academics: marks and grades given as percent of total marks.
• Statistics: population percentages, survey results, or composition of budgets (pie charts).
• Food/health: nutritional values often shown as percentages of daily intake.

Tips: Always write the percentage as a fraction of 100 when calculating, keep units (₹, marks, etc.), and check whether the percent is taken on the original amount or a changed amount (important for successive changes).

What this topic covers
Comparing quantities in Class 7 means measuring how one quantity relates to another using ratios, percentages, increases or decreases, discounts, profit and loss (and sometimes simple interest). The aim is to compare amounts quickly and accurately using standard shortcuts and methods.

Core ideas and quick rules

• Percent meaning: x% means x out of 100, so x% of a quantity A = (x/100) × A.
• Convert percent <> fraction <> decimal: x% = x/100 (fraction) = x/100 (decimal). Example: 12% = 12/100 = 0.12.
• Use multipliers for increase/decrease: For an increase of r%, new value = original × (1 + r/100). For a decrease of r%, new value = original × (1 − r/100).
• Successive percentage changes: Apply multipliers one after another (multiply the multipliers). Example: increase by 10% then decrease by 20% → total multiplier = 1.10 × 0.80.
• Reverse percentage (find original): If final value is known after r% change, original = final / (1 ± r/100) using + for decreases reversed and − for increases reversed appropriately.
• Discounts and marked price: Selling price = Marked price × (1 − discount%/100). For successive discounts multiply the remaining fractions.
• Profit and loss: Profit% or Loss% is always calculated on Cost Price (CP): Profit% = (Profit/CP)×100; Loss% = (Loss/CP)×100. Selling Price (SP) = CP × (1 + profit%/100) or CP × (1 − loss%/100).
• Unitary method & proportion: Convert to find 1% or 1 unit then scale up — useful when direct fraction is awkward.

Problem-solving shortcuts and tips

• Use 10% and 1% tricks: 10% of a number = divide by 10; 1% = divide by 100. Combine for other easy percentages (e.g., 15% = 10% + 5% = number/10 + number/20).
• For repeated changes use a single multiplier instead of recalculating each time: start × (1 ± r1/100) × (1 ± r2/100) ...
• When asked 'by what percent did it change?' use: % change = (difference / original) × 100. Always divide by the original (initial) quantity.
• To undo a discount or increase, divide by the corresponding multiplier: original = final / multiplier.
• Round intermediate steps only if safe; keep exact fractions or decimals until the final step to avoid errors.

Worked strategy outline (step-by-step)

1. Read carefully: identify whether percent is of which quantity (original, final, marked price, cost price).
2. Decide a method: multiplier method (fast) or unitary method (clearer for tricky wording).
3. Compute stepwise if successive changes occur (use multiplication of multipliers).
4. Check the answer by a quick estimate: does the final number make sense compared to the original?