Problems on Numbers Quiz Set 020

Let the numbers be $a$ and $10 - a$. We are given their product as a × (10 - a) = 9, which is a quadratic expression that can be simplified to $(a - 9) × (1 - a) = 0$. So the numbers could be 9 and 1.

We can see that $2{19/x}$ = ${17/5}$ × ${17/27}$. ⇒ ${2x + 19}/x$ = ${289/135}$ ⇒ x = 135.

Let the numbers be a and b. ab = 10a + b, and ba = 10b + a. The difference is 10a + b - (10b + a) = 9(a - b). It is given that 9(a - b) = 81, ⇒ a - b = $81/9$ = 9, ⇒ the difference between the digits is 9. If one of the digits is 4, the other cannot be determined.

Odd numbers are in an AP. First term $a = -1569$, common difference $d = 2$, the last term $t_n$ is given as 2425. By the AP formula, $2425 = -1569 + (n - 1) × 2$ ⇒ $n = 1 + {{2425 - (-1569)}/2} = 1998$.

Let the numbers be n - 1, n and n + 1. The sum is 3n. We are given 3n = -96, ⇒ n = $-96/3$, i.e., n = -32. So the smallest is -33.