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### Warm-up

### The Problem:

Jean's mean score is 22% less than her score in biology, her biology score was 10% higher than her chemistry, her English was the same as her mathematics result, and physics was half the mathematics score and 30% points lower than the mean.

What were Jean's scores in each subject and the mean score?

-/|\-

### Solution:

Let's work each statement in the question one after the other.

"Jean's mean score is 22% less than her score in biology" implies

$$M=b-22$$where $M$ is her mean score and $b$ is her score in biology.

"her biology score was 10% higher than her chemistry" implies

$$b=c+10$$where $c$ is her chemistry score.

"her English was the same as her mathematics result" implies

$$e=m$$where $e$ is English and $m$ is maths.

"and physics was half the mathematics score and 30% points lower than the mean" gives two mathematical statements

$$p=\frac{m}{2}$$and

$$p=M-30$$where $p$ is her physics score.

Now, let's consider the mean score ...

$$M=\frac{m+e+b+c+p}{5}$$Combining this with others above,

$$5(p+30)=2p+2p+M+22+b-10+p$$ $$5p+150=5p+M+12+b$$ $$138=M+b$$ $$138=b-22+b$$ $$b=80$$ $$M=b-22\Rightarrow M=48$$ $$b=c+10\Rightarrow c=70$$ $$p=M-30\Rightarrow p=18$$ $$p=\frac{m}{2}\Rightarrow m=e=36$$- So her scores are ...
- English 36%
- Mathematics 36%
- Biology 80%
- Chemistry 70%
- Physics 18%
- Mean score 48%

-oOo-

### Further Reading:

simultaneous equations

(care of mmerevise.co.uk)

Simultaneous equations - practice questions

(care of corbettmaths.com)