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Unitary Method

Competitive Mathematics · Arithmetic

6 Part4 worked examples14 practice questions

A shopkeeper says “12 bananas for ₹60”. How much for 20? Your mind first finds the price of one banana — ₹5 — and then multiplies by 20: ₹100. That is the unitary method: find the value of one unit, then multiply by the number asked for. It is the foundation of the work, speed and wages chapters that follow.

1

Direct proportion

One goes up, so does the other

When two quantities rise and fall together: number ↔ cost, days ↔ distance.

Value of one, then multiply

Quantity ∝ cost

Cost from quantity, quantity from cost, and converting units like dozens.

The direct proportion rule

Two quantities are in direct proportion if their ratio stays fixed. If 15 mangoes cost ₹120, 1 mango costs ₹8 and 25 mangoes cost ₹200. Cost–quantity–distance questions all fit this mould.

Example 1 · Basic
18 notebooks cost ₹270. What do 30 notebooks cost?
Solution, step by step
  1. 11 notebook .
  2. 230 notebooks
Answer ₹450
Example 2 · Standard
A mason lays 85 bricks a day. How many bricks in 3 weeks, and how many days for 9,350 bricks?
Solution, step by step
  1. 13 weeks days ⟹ bricks.
  2. 2For 9,350 bricks days.
Answer 1,785 bricks; 110 days
Example 3 · Basic
If 5 kg of mangoes cost ₹450, how many kg does ₹1,260 buy?
Solution, step by step
  1. 11 kg .
  2. 2₹1,260 buys kg.
Answer 14 kg
Example 4 · Standard
A factory makes 120 fans a day. How many in 3 weeks, and how many days for 5,040 fans?
Solution, step by step
  1. 13 weeks days ⟹ .
  2. 2For 5,040: days.
Answer 2,520 fans; 42 days
BasicTry it yourself first
8 pens cost ₹56. What will 12 pens cost?
BasicTry it yourself first
15 oranges cost ₹90. What will 25 oranges cost?
StandardTry it yourself first
24 notebooks cost ₹132. What will 40 notebooks cost?
StandardTry it yourself first
36 bulbs cost ₹270. What will 50 bulbs cost?
AdvancedTry it yourself first
45 mangoes cost ₹202.5. What will 72 mangoes cost?
BasicTry it yourself first
15 eggs cost ₹75. What do 4 dozen eggs cost?
StandardTry it yourself first
24 eggs cost ₹144. What do 5 dozen eggs cost?
AdvancedTry it yourself first
18 eggs cost ₹117. What do 6 dozen eggs cost?
BasicTry it yourself first
5 kg of sugar cost ₹200. How much sugar does ₹800 buy?
StandardTry it yourself first
12 kg of sugar cost ₹540. How much sugar does ₹2,250 buy?
AdvancedTry it yourself first
18 kg of sugar cost ₹1,260. How much sugar does ₹3,500 buy?
BasicTry it yourself first
A uniform rod of 20 m weighs 80 kg. What does 7 m of it weigh?
StandardTry it yourself first
A uniform rod of 45 m weighs 171 kg. What does 12 m of it weigh?
AdvancedTry it yourself first
A uniform rod of 36 m weighs 99 kg. What does 28 m of it weigh?
2

Inverse proportion

One goes up, the other comes down

Men ↔ days, speed ↔ time, ration ↔ people — the product stays fixed.

The product is fixed

Men × days = total work

Speed–time, men–days and ration questions.

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3

The chain rule

Several quantities change together

Men, days, hours and work — all in one balance.

Men × days × hours ÷ work

Equal on both sides

The link between two situations and wages questions.

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4

Traps and fast tricks

Slips · tricks

Where marks go in the unitary method.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Ratios and cancelling

Habits that shorten the work.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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