Go down to one, then up to what you want

Work out the value for a single unit, then multiply up. If 4 pens cost 60 rupees, one pen costs 15, so 7 pens cost 105.

Why it is worth learning

A ratio can be set up backwards without you noticing. The unitary method is harder to reverse, because the middle step has a meaning you can say out loud.

The direction of the first step tells you everything

Direct: 1 pen costs 60 ÷ 4. Inverse: 1 worker takes 6 × 10 = 60 days. Ask whether one worker should be faster or slower than six.

Situation	Step down to one	Step up to what you want
4 pens cost Rs 60. Find 7 pens	1 pen: 60 ÷ 4 = Rs 15	7 pens: 7 × 15 = Rs 105
6 workers take 10 days. Find 15	1 worker: 6 × 10 = 60 days	15 workers: 60 ÷ 15 = 4 days

Say the middle step out loud

Write the middle step as a sentence, not a bare number. One pen costing 240 rupees when four cost 60 sounds absurd, and hearing that is the check.

Problems with two changes at once

Handle one change at a time. More workers shrinks the days; more walls grows them.

Check the answer against common sense

Ask whether the answer should be larger or smaller than the number you started with, and check that it is.

Not every quantity scales

Ten machines fill ten times as many bottles, but ten people do not walk ten times as far. Read the situation before reaching for the arithmetic.

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Worked Example 1

If 12 exercise books cost Rs 540, what would 7 books cost?

Direct or inverse?Fewer books, less money, so direct

Step down to one1 book: 540 ÷ 12 = Rs 45

Step up to seven7 books: 7 × 45 = Rs 315

Sense check315 is less than 540, as it should be.

Worked Example 2

A tank is filled by 8 taps in 6 hours. How long would 12 taps take?

Direct or inverse?More taps, less time, so inverse

Step down to one tap1 tap: 8 × 6 = 48 hours

Step up to twelve taps12 taps: 48 ÷ 12 = 4 hours

Sense checkMore taps, fewer hours. An answer of 9 hours would have meant reversed steps.

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Activity 19.1

The price per unit hunt

What you needPrices and pack sizes for the same product in different sizes.

MethodWork out the price for one kilogram or one litre, using the same unit for every pack.

Then decideWhich pack size gives the best value? Is the largest pack always cheapest per unit?

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The unitary method is two steps: go down to ONE, then up to what you want.

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1What is the unitary method, and why is it safer than setting up a ratio?

2Why is the first step a division for pens but a multiplication for workers?

3How do you handle a problem where two things change at once?

4Why does the unitary method not apply to every situation?

5How can the unitary method help you decide which pack size is the best value?

Question 1 of 10

1The unitary method works by

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Unit 19 · Unitary Method

Unitary Method

Mathematics · Grade 7

What You Will Be Able To Do

1️⃣

Solve a proportion problem in two clear steps, through one unit

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Know whether the first step is a divide or a multiply, and say why

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Handle problems where two things change at the same time

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Work out which pack size in a shop is genuinely the best value

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The unitary method is two steps and nothing more. Work out the value for a single unit, then multiply up to however many units the question actually asks about.

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Presenter notes: Open by reminding the class of something they already did. Two units ago, when checking whether seven pens cost 105 rupees, somebody worked out the cost of one pen first and then multiplied. That side route has a name and it turns out to be the safest method in the whole of proportion work. Today it gets used deliberately rather than as a check.