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Smallest Fraction

Age 11 to 14 Short
Challenge Level

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


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