Entrance exam
Dean finishes his exam strongly. Can you work out how many questions are on the paper if he gets an average of 80%?
Age
14 to 16
Challenge level
Problem
In a university admissions test, Dean gets exactly 10 of the first 15 questions correct.
He then answers all the remaining questions correctly.
Dean finds out he has answered 80% of all the questions correctly.
How many questions are there on the test?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: 25
Using trial and improvement
Beginning: $\dfrac{10}{15}$
End: A fraction equivalent to $\dfrac45$
Numerator should be $5$ more than the denominator
$\dfrac 45 = \dfrac{12}{15} = \dfrac{16}{20} = \dfrac{20}{25}$ which has numerator $5$ more than denominator
$\dfrac{10}{15}$ plus $10$ more correct - total $25$ questions
Using the number of questions Dean got wrong
First $15$ questions: $5$ wrong
Other questions: $0$ wrong
Total $80\%$ correct $\therefore 20\%$ wrong
$5$ questions $=20\%$
$25$ questions $=100\%$
Using algebra
Suppose there are $n$ questions after the first $15$. Then Dean got $10+n$ right out of a total of $15+n$, so $\dfrac{10+n}{15+n}=80\%$.
$80\%$ is equivalent to $\dfrac{80}{100}=\dfrac{4}{5}$, so $$\begin{align}\frac{10+n}{15+n}&=\frac45\\
\Rightarrow10+n&=\frac45\times(15+n)\\
\Rightarrow(10+n)\times5&=4\times(15+n)\\
\Rightarrow50+5n&=60+4n\\
\Rightarrow50+n&=60\\
\Rightarrow n&=10\end{align}$$ So altogether there are $15+10=25$ questions on the test.