seegongsik
Grade 6 (age 11-12)

grade 6 Proportion & Proportional Distribution

A proportion is an equation that joins two equal ratios with an equals sign, like 2 : 3 = 4 : 6. The two outer numbers are called the extremes and the two inner numbers the means, and the product of the extremes always equals the product of the means. Sharing in a ratio means dividing a whole in a given ratio: 40 sweets shared 3 : 5 become 15 and 25. In the first picture, choose a, b and c and the fourth term is set so that it makes a proportion and the balance stays level; the second shows a whole shared in two parts as a pie chart.

What is a proportion?

⚖️ Two equal ratios joined by =
①2 : 3 and 4 : 6 have the same value, 2/3, so you can write 2 : 3 = 4 : 6
②The outer 2 and 6 are the extremes and the inner 3 and 4 are the means, and 2 × 6 = 12 equals 3 × 4 = 12
③When you change a, b and c in the picture, the fourth term is set so that a × □ = b × c, which keeps the balance level
④If the fourth term does not come out whole, it appears as a fraction: in 3 : 1 = 1 : □, □ = 1/3
2
3
4

Sharing in a ratio

🍰 Sharing a whole in a ratio
①To share 40 sweets between two brothers in the ratio 3 : 5, first think of the whole as 3 + 5 = 8 equal parts
②One part is 40 ÷ 8 = 5, so the older one gets 5 × 3 = 15 and the younger 5 × 5 = 25
③Check that the shares add back up to the whole: 15 + 25 = 40
④In the pie chart, A's slice is 3/8 of the whole and B's is 5/8
3
5
40

Summary

Property of a proportion
a : b = c : d → a × d = b × c
The product of the extremes equals the product of the means
Sharing in a ratio
A's share = Total × aa+b
A's share when a whole is divided in the ratio a : b
🎯 At a glance
①Proportion: two equal ratios joined by =
②Use extremes = means to find a missing term
③Sharing in a ratio: dividing a whole in a given ratio
④Check that the shares add up to the whole
⑤Used for scaling recipes, reading map scales and sharing pocket money

Examples & Unit Review

Example 1
Find □ in 3 : 4 = 9 : □.
1
The product of the extremes equals the product of the means: 3 × □ = 4 × 9.
2
3 × □ = 36, so □ = 36 ÷ 3 = 12.
▸ □ = 12
Set a = 3, b = 4 and c = 9 in the picture and the fourth term becomes 12.
Example 2
40 sweets are shared between an older and a younger brother in the ratio 3 : 5. How many does each get?
1
Think of the whole as 3 + 5 = 8 parts; one part is 40 ÷ 8 = 5.
2
The older brother gets 5 × 3 = 15 and the younger 5 × 5 = 25; check: 15 + 25 = 40.
▸ Older 15, younger 25
This is the starting setting of the second picture, whose pie chart shows A: 15 and B: 25.
Unit Review
In 2 : 5 = □ : 20, what is □?
①8
②10
③50
④4
▸ ① 8
1
Extremes = means: 2 × 20 = 5 × □.
2
5 × □ = 40, so □ = 40 ÷ 5 = 8.
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Ratio & Rate
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Percentage

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