Data Sufficiency

Of the \(48\) students in a class, \(16\) like to study History. What percentage of the girls in the class do not like to study History?

(1) One-third of the boys in the class like to study History.

(2) The number of girls who like History is \(50\) percent of the number of girls who do not like History.

Answer: D

Source: e-GMAT

Total number of students = 48
Students who like history = 16
Students who do not like history = 48 - 16 = 32

Target question => What percentage of the girls in the class do not like to study history?
$$i.e\ \frac{what\ is\ the\ number\ of\ girls\ who\ do\ not\ like\ history\ }{Total\ number\ of\ girls}\cdot\frac{100}{1}$$
Let the number of girls who do not like history = x
Let the total number of girls = y
Girls who like history = y - x
Total number of boys = 48 - y
Boys who like history = 16 - (y - x) = 16 - y + x
Boys who do not like history = 32 - x

$$Statement\ 1=>\ one-third\ of\ the\ boys\ in\ class\ like\ to\ study\ history$$
$$16-y+x=\frac{1}{3}\cdot\left(48-y\right)$$
$$16-y+x=\frac{48}{3}-\frac{y}{3}$$
$$16-y+x=16-\frac{y}{3}$$
$$-y+x=\frac{-y}{3}$$
$$x=\frac{-y}{3}+\frac{y}{1}=\frac{-y+3y}{3}=\frac{2y}{3}$$
$$\exp res\sin g\ it\ in\ terms\ of\ \frac{x}{y};\ divide\ through\ by\ y$$
$$x\div y=\frac{2y}{3}\div y$$
$$x\cdot\frac{1}{y}=\frac{2y}{3}\cdot\frac{1}{y}$$
$$\frac{x}{y}=\frac{2}{3}$$
$$\%\ of\ girls\ who\ don't\ like\ history\ =\ \frac{2}{3}\cdot100\ =\ 67\%$$
$$Statement\ 1\ is\ SUFFICIENT$$


$$Statement\ 2\ =>\ The\ number\ of\ girls\ who\ like\ history\ is\ 50\%\ of\ the\ number\ of\ girls\ who\ do\ not\ like\ history$$
$$y-x=50\%\ of\ x$$
$$y-x=\frac{50}{100}\cdot x$$
$$y-x=\frac{x}{2}$$
$$2\left(y-x\right)=x$$
$$\left(\exp res\sin g\ it\ in\ terms\ of\ \frac{x}{y}\right)$$
$$2y-2x=x$$
$$2y=x+2x$$
$$\frac{2y}{y}=\frac{3x}{y}$$
$$2=\frac{3x}{y}$$
$$2\div3=\frac{3x}{y}\div3$$
$$2\cdot\frac{1}{3}=\frac{3x}{y}\cdot\frac{1}{3}$$
$$\frac{2}{3}=\frac{x}{y}$$
$$\%\ of\ girls\ who\ don't\ like\ history=\frac{2}{3}\cdot100=67\%$$
$$statement\ 2\ is\ SUFFICIENT$$

$$Since\ each\ statement\ alone\ is\ SUFFICIENT,$$
$$Answer\ =\ D$$