Questions based on direct proportionality

Two quantities are directly proportional when they increase and decrease together at a steady rate.
If one doubles, the other doubles; if one is halved, the other is halved.
For example, if 1 pen costs $5, then 5 pens cost $25 and 10 pens cost $50.

The graph of a direct proportion is a straight line through the origin.
As one quantity increases, the other increases in step, keeping their ratio constant.
When one quantity is zero, so is the other.

To solve, put the two situations on separate rows, then cross-multiply across them. Cross-multiplying gives $1 \times X = 5 \times 5$, so $X = 25$, meaning 5 pens cost $25.

The first row does not have to start with 1. Cross-multiplying gives $3 \times X = 15 \times 7 = 105$, so $X = \frac{105}{3} = 35$.
The method is the same whatever the starting number.

If 1 apple costs $2, how much do 6 apples cost? Cross-multiply: $1 \times X = 2 \times 6$, so $X = 12$ dollars.

If 3 notebooks cost $18, how much do 7 notebooks cost? Cross-multiply: $3 \times X = 18 \times 7 = 126$, so $X = \frac{126}{3} = 42$ dollars.

If 2 grams of paint cover 1 square cm, how much paint covers 616 square cm? Cross-multiply: $1 \times X = 2 \times 616$, so $X = 1232$ grams.