You may also like

problem icon

2-digit Square

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

problem icon

Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

problem icon

Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Length, Width and Area

Age 14 to 16 Short Challenge Level:

Answer: 9


Suppose the width is $w$, then the length is $w+16$.

The area is equal to the product of the length and the width:
Area is $w\times(w+16) = 225$.

So the length and the width are a factor pair of 225.

225 = 5 $\times$ 5 $\times$ 3 $\times$ 3
225 = 5 $\times$ 45,     difference = 40
225 = 15 $\times$ 15,   difference = 0
225 = 9 $\times$ 25,     difference = 16
$\therefore$ width = 9


Using trial & improvement

We can try different numbers for $w$ (knowing that it is 16 less than the length):

If $w=1$, then the area would be $1\times 17=17$ - too small.
If $w=20$, then the area would be $20\times 36=720$ - too big.
If $w=5$, then the area would be $5\times 21=105$ - too small.
If $w=10$, then the area would be $10\times 26=260$ - just too big.
If $w=9$, then the area would be $9\times 25=225$ - perfect.

So $w=9$. 


Using algebra

Area $w^2+16w=225$

Completing the square
$(w+8)^2=w^2+8w+8w+64=(w^2+16w)+64$

So $(w+8)^2-64 = w^2+16w\\
\begin{align}\therefore w^2+16w=225&\Rightarrow(w+8)^2-64 =225\\
&\Rightarrow (w+8)^2=289\\
&\Rightarrow (w+8)=\pm17\\
&\Rightarrow w=\pm17 - 8\end{align}$
$w$ must be positive, so $w=17-8=9$.


Using the quadratic formula
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

$w$ is the variable $x$, and $a=1$, $b=16$, $c=-225$

$\begin{align}w&=\dfrac{-16\pm\sqrt{16^2+4\times225}}{2}\\
&=-8\pm\sqrt{16\times4+225}\\
&=-8\pm\sqrt{289}\\
&=-8\pm17\end{align}$
Need width to be positive so $w=-8+17=9$





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