Problems on Numbers Quiz Set 011

Let the numbers be a and b, and let their ratio be k such that $a/b = k$. We are given $a + b = 53$ ⇒ $b(k + 1) = 53$. Similarly, from the difference we can obtain $b(k - 1) = 19$. Dividing we get ${k + 1}/{k - 1} = 53/19$. By componendo and dividendo, $k = {53 + 19}/{53 - 19}$ = ${36/17}$, which is same as: $2{2/17}$.

Let the numbers be $a$ and $30 - a$. We are given their product as a × (30 - a) = 161, which is a quadratic expression that can be simplified to $(a - 23) × (7 - a) = 0$. So the numbers could be 23 and 7.

If the above number has to be divisible by 3, the sum of the digits, i.e., 3 + 9 + 2 + 4 + 2 + x + 6 + 4 + 1 + 6, should be divisible by 3. So we can see that $x + 37$ should be divisible by 3. By inspection, x = 2.

Let the numbers be 2n - 3, 2n - 1 and 2n + 1. The sum is 3(2n - 1) = 3 x middle. We are given 3 x middle = 657, ⇒ middle = $657/3$, i.e., middle = 219.

The first term is 70, common difference is d = -7, n-th term is 0. So $0 = 70 + (n - 1) × -7$ which gives $n = 1 + {70/7} = 11$.