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