# Math Puzzle | Find Out The Numbers Inside The Boxes

Find the numbers inside the boxes

## Solutions – Finding the numbers inside the square

Replace question marks with a, b, c and d

From figure

ab = 12…………..eq(1)

a + c = 5…………eq(2)

cd = 10…………..eq(3)

b + d = 9…………eq(4)

From eq(3) & eq(4)

cd = 10

**⇒** c(9 – b) = 10

From eq(2)

c = 5 – a

**⇒** (5 – a)(9 – b) = 10

From eq(1)

a = 12/b

**⇒** (5 – 12/b)(9 – b) = 10

Multiply with b

**⇒** (5b – 12)(9 – b) = 10b

Expand the equation, so

45b – 5b² – 108 + 12b = 10b

**⇒** 5b² – 47b + 108 = 0

## This is a quadratic equation in b, so apply the quadratic formula to find the value of b

\begin{aligned} b=&\frac{47\pm \sqrt{(-47)^2-4 \times5\times108}}{2\times5} \\ \\ =&\frac{47\pm \sqrt{49}}{10} \\ \\ =&\frac{47\pm 7}{10} \\ \\ \Rightarrow b=&4 \ \ \& \ \ b = \frac{27}{5} \\ \end{aligned}

We know a, b, c and d are the numbers inside the boxes, so we can use the value of b to find the values od a, c and d

b has two values, so in the 1st case we start with b = 4, then

d = 9 – b = 9 – 4 = 5

a = 12/b = 12/4 = 3

c = 5 – a = 5 – 3 = 2

so, a = 3, b = 4, c = 2 & d = 5

2nd case b = 27/5

d = 9 – b = 9 – 27/5 = 18/5

a = 12/b =12/(27/5) = 20/9

c = 5 – a = 5 – 20/9 = 25/9

so, a = 20/9, b = 27/5, c = 25/9 & d = 18/5