Mr. and Mrs. Ahuja weigh $x$ and $y$ $kg$. Find their present weights.

Question

Mr. and Mrs. Ahuja weigh $x$ kg and $y$ kg respectively. They both take a dieting course at the end of which Mr. Ahuja loses $5$ kg and weighs as much as the wife weighed before the course. Mrs. Ahuja loses $4$ kg and weighs $7$/$8$th of what her husband weighed before the course. From two equations in $x$ and $y$ and hence find their present weights.

I tried the following,

Mr. Ahuja's weight before the course$=$$x$ kg

Mrs. Ahuja's weight before the course$=$$y$ kg

                              After dieting course

Mr. Ahuja's weight: $E_1 =>x-5=y$

Mrs. Ahuja's weight: $E_2 =>y-4=\frac78 x$

Solving for $x$, I am getting $-27$ which is not possible.

Where did I go wrong?

Answer 1

$$y = x-5 \;\text{and}\; y - 4 = \frac 78 x \implies y - 4 = \underbrace{(x-5)}_{\large y} - 4 = \frac 78 x$$

Multiplying both sides of the equation by $8$ gives us $$\begin{align} 8(x-5) - 8\cdot 4 = 7x & \iff 8x - 72 -7x = 0 \\ &\iff x = 72\text{ kg}.\end{align}$$

Now solve for $y = x - 5 = 72-5 = 67\;\text{ kg}$.

Recall that $x, y$ give the weights prior to losing weight. So we need to find current weights: Mr: $x- 5 = 72 - 5 = 67$, Mrs: $67 - 4 = 63$.

Answer 2

Hint:

$y=x-5$

$y=\frac78 x +4$

Set them equal to each other, solve for $x$. Then substitute $x$ into one of the original equations to find $y$.

Should find that $x=72$ kg and $y=67$ kg.