Bradley's Maths
Combining Ratios
GCSE all examination boards and IGCSE Cambridge (0580)
Exam Question
Rose, Sarah and Tabitha are sisters.
The ratio of Rose's age to Sarah's age is $3:2$
The ratio of Sarah's age to Tabitha's age is $3:5$
a) Find the ratio of Rose's age to Tabitha's age.
b) Given that their combined age is 100, find the age of each of them
Before you move on to the solution or watch the video, why don't you have a go at this one yourself?
Solution
The difficulty with this question is, although Sarah is common to both parts of the given information, the number representing her age is different in each item of information.
We need each ratio to have the same value for Sarah's age and to get that we use the same method we use when eliminating a variable in simultaneous equations.
We multiply the first ratio by 3 and the second ratio by 2 to get:
$$\text{Rose }:\text{ Sarah}=3(3:2)=9:6$$ $$\text{Sarah }:\text{ Tabitha}=2(3:5)=6:10$$So the ratio of the three sisters is $9:6:10$
And the answer to part a) is:
$$\text{Rose }:\text{ Tabitha}=9:10$$Part b) is now just a straightforward ratio question:
\begin{align*} 9+6+10 &= 25\\ \frac{100}{25} &= 4\\ 4\times9 &= 36\\ 4\times6 &= 24\\ 4\times10 &= 40 \end{align*}Final answer:
$$\textbf{Rose is 36, Sarah is 24, Tabitha is 40}$$Check: Their combined ages $36+24+40=100$ — so our answer is sensible.
If a ratio is given as $a:b_1$ and $b_2:c$ where $b_1$ and $b_2$ represent the same item then the way to combine them is to multiply the entire ratio $a:b_1$ by $b_2$ and the entire $b_2:c$ by $b_1$ so that we have $ab_2:b_1b_2:b_1c$
Is there a topic you'd like to see featured? Request it on my Q & A page and I'll prioritize it for a future post.
Topics covered in this question
- Combining ratios
- Using ratios to find unknown values