### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Fraction of Percentages

##### Age 14 to 16 ShortChallenge Level

Answer: $\frac3{16}$

Using fractions
$W$ is $25\%$ of $X$, which is $\frac14$ of $X$

$X$ is $45\%$ of $Y$, so $X=\frac9{20}Y$

$Z$ is $60\%$ of $Y$, so $Z=\frac35Y$
\begin{align} W&=\tfrac14X\\ &=\tfrac14\left(\tfrac9{20}Y\right)\\ &=\tfrac9{80}Y\end{align}
$Z=\frac35Y$ and so $Y=\frac53Z$
\begin{align} \tfrac9{80}Y&=\tfrac9{80}\left(\tfrac53Z\right)\\ &=\tfrac3{16}Z\end{align}

Using $100$ parts of $Y$
Suppose $Y$ is split into $100$ parts, so that each part is $1\%$ of $Y$.
Then $Z$ is $60\%$ of $Y$ so $Z$ is $60$ parts, and $X$ is $45$ parts.

So $50\%$ of $X$ would be half of $45$ parts, which is $22.5$ parts, and $25\%$ of $X$, which is $W$, is half of $22.5$ parts. Half of $22.5$ is $11.25$, so $W=11.25$ parts.

Then $\frac{W}{Z}=\frac{11.25}{60}$

Multiplying the numerator and denominator by $4$ to get whole numbers gives $\frac{W}{Z}=\frac{45}{240}$ which simplifies $\frac{45}{240}=\frac{9}{48}=\frac{3}{16}$.

Using a diagram (and fractions)
$60\%$ and $45\%$ can both be easily represented by splitting the whole into $20$ pieces, as shown in the diagram below. Beginning by splitting $Y$ into $20$ pieces, $Z$ is $60\%$ of $Y$ so $12$ pieces (since $60\%$ is the same as $\frac6{10}$ or $\frac{12}{20}$) and $X$ is $45\%$ of $Y$ so $9$ pieces (since $45\%$ is halfway between $\frac4{10}$ and $\frac5{10}$, so halfway between $\frac8{20}$ and $\frac{10}{20}$, which is $\frac9{20}$).

$W$ is $25\%$ of $X$, which is the same as $\frac14$ of $X$. $X$ is made up of $9$ pieces, so it is difficult to find $\frac{1}{4}$ of $X$ as $9$ is not divisible by $4$.

By splitting each of the $9$ pieces into $4$ smaller pieces, $X$ will be represented as $9\times4=36$ smaller pieces, and $W$ is the same as $9$ of these smaller pieces, as shown below.

To compare $Z$ and $W$, we should also split each of the $12$ pieces of $Z$ into $4$ smaller pieces, so that the pieces in $Z$ are the same size as the pieces in $W$. That means $Z$ will be made up of $12\times4=48$ small pieces, as shown at the bottom of the diagram.

So $\frac{W}{Z}=\frac{9}{48}$, which simplifies to $\frac{3}{16}$ (by dividing numerator and denominator by $3$).

Using percentages
$W$ is $25\%$ of $X$, which is $45\%$ of $Y$, so $W$ is $25\%$ of $45\%$ of $Y$.

$Z$ is $60\%$ of $Y$. So $\frac{W}{Z}=\dfrac{25\%\times45\%\times Y}{60\%\times Y}=\dfrac{25\%\times45\%}{60\%}$.

Writing the percentages as fractions over $100$, this is equal to $$\begin{split}\frac{\frac{25}{100}\times\frac{45}{100}}{\frac{60}{100}}&=\frac{25\times\frac{45}{100}}{60}\\ \hspace{1mm}\\ &=\frac{25\times45}{60\times100}\\ \hspace{1mm}\\ &=\frac{25\times9\times5}{12\times5\times 25\times4}\\ \hspace{1mm}\\ &=\frac{9}{12\times4}\\ \hspace{1mm}\\ &=\frac{3\times3}{3\times4\times4}\\ \hspace{1mm}\\ &=\frac{3}{16}\end{split}$$

You can find more short problems, arranged by curriculum topic, in our short problems collection.