### Off the Cuff

Can you work out the ratio of shirt types made by a factory, if you know the ratio of button types used?

### A Leg to Stand On

Can you work out the number of chairs at a cafe from the number of legs?

### Out of Sync

Albert Einstein could see two clocks which were out of sync. For what fraction of the day did they show the same time?

# Brothers and Sisters

##### Age 11 to 14 ShortChallenge Level

Answer: 4 brothers and 3 sisters

Working it out starting from small numbers

Start from the brother - he must have at least one brother and one sister.

What about the sister? Add another sister and brother so that she has twice as many brothers as sisters.

Need to add another brother so that the brother has the same number of brothers and sisters

Need to add more so that the sister has twice as many brothers as sisters

This works for the brother too.

Using algebra
Let $b$ represent the number of brothers in the family and $s$ represent the number of sisters in the family.

Each brother has $b-1$ brothers and $s$ sisters.
Each sister has $b$ brothers and $s-1$ sisters.

The boy has the same number of brothers as sisters, so $b-1=s$.
Each sister has half as many sisters as brothers, so $s-1=\dfrac{1}{2}b$.

Solving by substitution or elimination (see below), $b=4$ and $s=3$, so there are $7$ siblings in total.

Solving by substitution
$s-1=\dfrac{1}{2}b$, so $2s-2=b$ (by multiplying by 2).

Substituting $b=2s-2$ into $b-1=s$ gives $2s-2-1=s$, so $2s-3=s$, so $s=3$.

Substituting $s=3$ into $b-1=s$ gives $b=4$.

Therefore there are $7$ siblings in total.

Solving by elimination
Subtracting $s-1=\dfrac{1}{2}b$ from $s=b-1$ gives $s-(s-1)=b-1-\left(\dfrac{1}{2}b\right)$.
This simplifies to $s-s+1=b-\dfrac{1}{2}b-1$, so $1=\dfrac{1}{2}b-1$.
Adding 1 to both sides, $2=\dfrac{1}{2}b$, and multiplying by 2 gives $4=b$.

Substituting $b=4$ into $b-1=s$ gives $s=3$.

Therefore there are $7$ siblings in total.

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