Concepts: Proportional Reasoning-1 (part 1)

1.1 Similar shapes and proportional change

Digital images can be enlarged or shrunk. Look at five versions of the same tiger photograph in the chapter. Images A, C and D look alike; B looks stretched and E looks compressed. Measurement explains why.

Compare A with C. The width of C is half the width of A, and the height of C is also half the height of A. Both dimensions changed by the same factor, one half, so the picture still looks like a tiger.

Compare A with B. The width and the height both fell by 20 mm, but the width did not fall by the same factor as the height: the height halved while the width did not. Equal differences do not protect the shape. Compare A with D: both dimensions grew by the same factor 3/2, so A, C and D are similar.

We say the changes are proportional when the widths and heights change by the same multiplicative factor. That single idea runs through the whole chapter.

1.2 Ratios and their terms

A ratio compares two quantities of the same kind. For image A write the width-to-height comparison as 60 : 40. The two numbers are called the terms of the ratio. Image C gives 30 : 20 and image D gives 90 : 60.

A ratio a : b says that the first quantity contributes a units for every b units of the second. Image A, for instance, has 40 mm of height against each 60 mm of width.

Ratios compare quantities with the same unit; once the unit is the same, the ratio is just a pair of numbers. The order of the terms matters: width : height is not the same as height : width. When the units differ, convert to a common unit before writing the ratio, a habit that returns in the unit-conversion section.

1.3 Ratio, fraction and rate

A ratio is not the same as a fraction of the whole. The ratio 3 : 2 can describe a rectangle with width 3 units and height 2 units; the same numbers as a fraction, 3/2, only tell you the width when the height is 1. A ratio keeps two quantities side by side, while a fraction compares a part with a whole.

Ratios also differ from rates because the two quantities are of the same kind and the same unit. Width : height, boys : girls and coffee : milk are ratios. A rate such as kilometres per hour compares different kinds, and it needs its own unit. Keeping these three ideas apart stops a common error: writing a ratio when a rate is needed, or treating a ratio as a fraction of the total. When a question asks "what fraction of the whole", add the terms first; when it asks "in what ratio", keep the terms separate.

1.4 Simplest form and proportion

Dividing both terms by their HCF gives the simplest form of a ratio. The HCF of 60 and 40 is 20, so 60 : 40 = 3 : 2. For image D, divide 90 and 60 by 30 to get 3 : 2 as well. Image B gives 40 : 20 = 2 : 1, and image E gives 60 : 60 = 1 : 1. The simplest forms of B and E are different from 3 : 2, which is why those images do not look like the others.

When two ratios have the same simplest form, they are in proportion. The symbol :: is read as "is proportional to" or "is as". The order still matters, and both pairs must compare the same two quantities in the same order, which is why A is proportional to both C and D.

1.5 Worked examples from the textbook

Example 1: are 3 : 4 and 72 : 96 proportional? The first ratio is already in simplest form. Dividing 72 and 96 by their HCF, which is 24, reduces the second ratio to 3 : 4. The simplest forms match, so the ratios are proportional.

Example 2: Kesang makes lemonade. She mixes 6 glasses with 10 spoons of sugar. For 18 glasses she needs the same sweetness, so the ratios must be proportional. The first term grows from 6 to 18, a factor of 3, and the second term must grow by the same factor: 10 × 3 = 30 spoons. So 6 : 10 :: 18 : 30.

Example 3: Nitin builds 60 feet of wall with 3 bags of cement, and Hari builds 40 feet with 2 bags. Nitin worries that Hari's wall is weaker because it used less cement. Compare the ratios of length to cement: 60 : 3 = 20 : 1 and 40 : 2 = 20 : 1. They are proportional, so the walls are equally strong.