Two frequencies are missing from this table of scores in a 200-mark scholarship test. There were 100 candidates in all and the median score is 152.
100-120 -> 12, 120-140 -> $x,140-160->25,160-180->y$, 180-200 -> 14.
Find $x$ and $y$.

The frequencies total 100: $12+x+25+y+14=100$, so $x+y=49$. With $\frac{n}{2}=50$ and a median of 152, the median class is 140-160, so $l=140$, cf = $12+x,f=25$ and $h=20$. Then $152=140+\left(\frac{50-(12+x)}{25}\right)\times 20$, i.e. $12=\frac{4(38-x)}{5}$, giving $38-x=15,x=23$ and $y=26$. Check: 12, 23, 25, 26, 14 has cumulative 12, 35, 60, 86, 100 and $140+\frac{50-35}{25}\times 20=152$. '$x=26,y=23$' reports the pair the wrong way round. '$x=8,y=41$' uses $h=10$ instead of the class width 20. '$x=24,y=25$' just splits 49 almost equally without ever using the median.