Question

Two brands of tyre X,Y were tested with following result. If combined mean is $31.7$,
find $a$.

life(miles)	X	Y
$20-25$	$1$	$0$
$25-30$	$22$	$24$
$30-35$	$64$	$76$
$35-40$	$a$	$0$
$40-45$	$3$	$0$

• A. $6$
• B. $10$
• C. $12$
• D. $4$

Answer 1

The correct option is B $10$

$Life\left(miles\right)$	$Class-mark$ $A$	$X$	$Y$	$AX$	$AY$
$20-25$	$22.5$	$1$	$0$	$22.5$	$0$
$25-30$	$27.5$	$22$	$24$	$605$	$660$
$30-35$	$32.5$	$64$	$76$	$2080$	$2470$
$35-40$	$37.5$	$a$	$0$	$37.5a$	$0$
$40-45$	$42.5$	$3$	$0$	$127.5$	$0$
		$\sum X=90+a$	$\sum Y=100$	$\sum AX=2835+37.5a$	$\sum AY=3130$

$\overline{X}=\frac{\sum AX}{\sum X}=\frac{2835+37.5a}{90+a}$

$\overline{Y}=\frac{\sum AY}{\sum Y}=\frac{3130}{100}=31.3$

$\Rightarrow$ Combined mean $=\frac{\sum X.\overline{X}+\sum Y.\overline{Y}}{\sum X+\sum Y}$

$\Rightarrow$ $31.7=\frac{(90+a).\frac{2835+37.5a}{90+a}+100\times 31.3}{90+a+100}$

$\Rightarrow$ $31.7=\frac{2835+37.5a+3130}{190+a}$

$\Rightarrow$ $31.7=\frac{5965+37.5a}{190+a}$

$\Rightarrow$ $6023+31.7a=5965+37.5a$

$\Rightarrow$ $58=5.8a$

$\therefore$ $a=10$