Fundamentals of Ratio Division

A ratio tells you how quantities compare. If £40 is divided between A and B in the ratio $3:5$, imagine A receiving 3 equal-sized parts and B receiving 5. There are $3+5=8$ equal parts altogether, so each part is worth $£40\div8=£5$. A receives $3\times£5=£15$ and B receives $5\times£5=£25$. The shares add back to £40.

In general, to divide a total in a part : part ratio, add the ratio numbers to find the number of equal parts. Divide the total by that number, then multiply by each ratio number to find the shares. The order matters: $3:5$ gives the first person 3 parts, not 5.

When one number represents the whole

A part : whole ratio works differently. Suppose the ratio of red counters to all counters is $3:8$, and there are 40 counters. The 8 represents the entire set, so one equal part is $40\div8=5$ counters. There are $3\times5=15$ red counters and $40-15=25$ counters that are not red. The ratio of red to non-red counters is therefore $3:5$. Adding 3 and 8 at the start would count the red counters twice, because they are already included in the whole.

Describing a division as a ratio

You can also work backwards from two known shares. If £18 goes to A and £24 to B, their part : part ratio is $18:24$. Divide both numbers by 6 to give $A:B=3:4$. The corresponding part : whole ratio for A is $18:(18+24)=18:42=3:7$. State which quantities the ratio compares, because $3:4$ and $3:7$ describe the same division in different ways.