# Smallest Fraction

Which of these is the smallest?

Which of these is the smallest?

$$\frac{2+3}{4+6}\hspace{30mm}\frac{2\div3}{4\div6}\hspace{30mm}\frac{23}{46}\hspace{30mm}\frac{2-3}{4-6}\hspace{30mm}\frac{2\times3}{4\times6}$$

*This problem is taken from the World Mathematics Championships*

**Answer**: $\dfrac{2\times3}{4\times6}$

**Finding the value of each one**

$\dfrac{2+3}{4+6}=\dfrac5{10}=\dfrac{1}{2}$

$\dfrac{2\div3}{4\div6}=\dfrac{\frac23}{\frac46}=\dfrac{\frac23}{\frac23}=1$

$\dfrac{23}{46}=\dfrac12\\$

$\dfrac{2-3}{4-6}=\dfrac{-1}{-2}=\dfrac12\\$

$\dfrac{2\times3}{4\times6}=\dfrac{6}{4\times6}=\dfrac14$

So $\dfrac{2\times3}{4\times6}$ is the smallest.

**Preserving the ratio between the top and bottom numbers**

All relate to $\frac24=\frac12$, numerator $:$ denominator $=1:2$

$\dfrac{2+3}{4+6}$

Numerator: $+2$

Denominator: $+4$

Operations are in the ratio $1:2$ so this fraction is still $\frac12$

$\dfrac{2\div3}{4\div6}$

Numerator: $\div3$

Denominator: $\div6$

Denominator gets smaller twice as quickly as numerator $\therefore$ fraction doubles in size

$\dfrac{23}{46}$, numerator $:$ denominator $=1:2$

$\dfrac{2-3}{4-6}$

Numerator: $-2$

Denominator: $-4$

Operations are in the ratio $1:2$ so this fraction is still $\frac12$

$\dfrac{2\times3}{4\times6}$

Numerator: $\times3$

Denominator: $\times6$

Denominator gets larger twice as quickly as numerator $\therefore$ fraction halves in size

$\therefore$ smallest