Quick questions on Ratio, rate and proportion explained: O-Level E-Maths

A ratio compares quantities of the same kind, written $a : b$. Simplify by dividing both parts by their highest common factor, so $18 : 24 = 3 : 4$. To share a quantity in a given ratio, add the parts to find the total number of parts, find the value of one part, then multiply.

A rate compares two quantities of different kinds, such as distance and time. The most common is speed:

Two quantities are in direct proportion when one is a constant multiple of the other, so doubling one doubles the other. We write $y \propto x$, meaning $y = kx$ for a constant $k$. The ratio $\dfrac{y}{x}$ stays fixed. Cost per item bought at a fixed price is a direct proportion.

Two quantities are in inverse proportion when their product is constant, so increasing one decreases the other in the same ratio. We write $y \propto \dfrac{1}{x}$, meaning $xy = k$. The number of workers and the time to finish a fixed job is an inverse proportion.

A ratio answer such as $6 : 9$ should be given in simplest form, $2 : 3$.

Simplify the ratio $45 : 60$. [1 mark]

A runner covers $4\ \text{km}$ in $25$ minutes. Find the average speed in kilometres per hour. [2 marks]

If $3$ workers build a wall in $12$ days, how long would $9$ workers take at the same rate? [2 marks]