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# Order the Products

**Answer**:

186$\times$214 (smallest)

210$\times$190

195$\times$205

198$\times$202

200$\times$200 (largest)

**Numerically, using 200$\times$200**

210$\times$190 = (200 + 10)(200 $-$ 10) = 200$^2$ + 2000 $-$ 2000 $-$ 100

= 200$^2-$ 100

195$\times$205 = (200 $-$ 5)(200 + 5) = 200$^2$ $-$ 1000 + 1000 $-$ 25

= 200$^2-$ 25

198$\times$202 = (200 + 2)(200 $-$ 2) = 200$^2-$ 4

186$\times$214 = (200 + 14)(200 $-$ 14) = 200$^2-$ 14$^2$

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

**Writing the numbers in terms of 200**

All of the products are $(200 +n)(200 -n)$ for some small value of $n$.

$\begin{align}(200+n)(200-n)&=200^2+200n-200n-n^2\\

&=200^2-n^2\end{align}$

$n$ larger $\Rightarrow200^2-n^2$ smaller (for $n\gt0$).

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

*Diagrammatic representation*

Red strip and green strip both have width $1$, area $200$

Red strip and green strip both have width $n$, area $200n$

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Age 14 to 16

ShortChallenge Level

- Problem
- Solutions

186$\times$214 (smallest)

210$\times$190

195$\times$205

198$\times$202

200$\times$200 (largest)

210$\times$190 = (200 + 10)(200 $-$ 10) = 200$^2$ + 2000 $-$ 2000 $-$ 100

= 200$^2-$ 100

195$\times$205 = (200 $-$ 5)(200 + 5) = 200$^2$ $-$ 1000 + 1000 $-$ 25

= 200$^2-$ 25

198$\times$202 = (200 + 2)(200 $-$ 2) = 200$^2-$ 4

186$\times$214 = (200 + 14)(200 $-$ 14) = 200$^2-$ 14$^2$

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

All of the products are $(200 +n)(200 -n)$ for some small value of $n$.

$\begin{align}(200+n)(200-n)&=200^2+200n-200n-n^2\\

&=200^2-n^2\end{align}$

$n$ larger $\Rightarrow200^2-n^2$ smaller (for $n\gt0$).

$\therefore$ 186$\times$214 $\lt$ 210$\times$190 $\lt$ 195$\times$205 $\lt$ 198$\times$202 $\lt$ 200$\times$200

Red strip and green strip both have width $1$, area $200$

Red strip and green strip both have width $n$, area $200n$

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

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?

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

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