Proportional Reasoning: Missing Values and Graphs

1. Finding missing values in a table

In a table of equivalent ratios, the two quantities are proportional: when one is multiplied by a number, the other is multiplied by the same number. Here is a table from the NaCCA curriculum with two missing values, x and y.

Method 1: scale factor down the table. Going from 3 to 6 we multiply by 2, so x = 10 × 2 = 20. Going from 10 to 40 we multiply by 4, so y = 3 × 4 = 12. Step through the animation to see this.

▶ Whatever you do to one number in a row, do the same to its partner. Watch the multipliers on both sides match.

Method 2: the multiplier across. In every row, second quantity = first quantity × $\frac{10}{3}$. So x = 6 × $\frac{10}{3}$ = $\frac{60}{3}$ = 20, and for y we go backwards: y = 40 ÷ $\frac{10}{3}$ = 40 × $\frac{3}{10}$ = 12.

Method 3: equal fractions. Write a proportion and make the fractions match: $\frac{3}{10}=\frac{6}{x}$. The top was multiplied by 2, so the bottom must be too: x = 20. This is the quickest method when one number is an obvious multiple of another.

2. Proportion in real life: scales and exchange rates

A map scale of 1 cm : 5 km means every centimetre on the map stands for 5 km on the ground. The unit rate is 5 km per cm, so real distance = 5 × map distance.

A forex bureau works the same way. Suppose the rate is GH₵15 for 1 US dollar (rates change every day; this is just for practice). Then cedis = 15 × dollars, and dollars = cedis ÷ 15.

3. Plotting proportional pairs on a graph

Each column of a table is a pair of numbers that we can plot as a point (x, y) on the coordinate plane. Exercise books at a bookshop in Ho cost GH₵2 each, so cost y = 2 × number of books x. Watch the table become a graph.

▶ Here x is the number of books and y is the cost in cedis. Notice the points form a straight line that starts at the origin (0, 0).

• The points lie on a straight line.
• The line passes through the origin (0, 0): zero books cost GH₵0.
• Each step of 1 to the right goes up by the same amount, 2: this is the unit rate.

4. Is it really proportional?

Not every table that "goes up steadily" is proportional. A taxi in Takoradi charges a fixed GH₵5 plus GH₵2 per kilometre:

The fare goes up by GH₵2 each time, but fare ÷ distance is not constant: $\frac{7}{1}\ne \frac{9}{2}$. Doubling the distance from 1 km to 2 km does not double the fare. On a graph the points still lie on a straight line, but it crosses the fare axis at 5, not at the origin. So the fare is not proportional to the distance.

Worked examples

Try each one yourself first, then open the solution and compare every step.