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Proportions (a : b = c : d)

Linear equations

Japanese school year: Junior high 1

What you learn

You will learn how to solve for an unknown in proportions using the property that the product of extremes equals the product of means. This skill is widely applied in calculating geometric scales, map ratios, and recipe mixtures. A solid understanding of basic ratios and linear equations is recommended beforehand.

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Key points

This fundamental property solves proportions by converting them into equations. The product of the outer terms adad (extremes) always equals the product of the inner terms bcbc (means).

a:b=c:d  ⟺  ad=bca : b = c : d \iff ad = bc

This formula expresses a proportion as an equation of equal ratio values. It indicates that ab=cd\frac{a}{b} = \frac{c}{d}, and cross-multiplying yields ad=bcad = bc.

a:b=c:d  ⟺  ab=cd(b≠0,  d≠0)a : b = c : d \iff \frac{a}{b} = \frac{c}{d} \quad (b \neq 0,\; d \neq 0)

This formula defines the "value of a ratio" a:ba : b as a number. Dividing the first term aa by the second term bb yields the fraction ab\frac{a}{b}, representing their relative proportion.

a:b=ab(b≠0)a : b = \frac{a}{b} \quad (b \neq 0)

When a term in a proportion is an expression like x+1x + 1, enclose it in parentheses as (x+1)(x + 1) before multiplying extremes and means, ensuring proper distribution.

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