Problems on Numbers Quiz Set 010

0.67676767... is same as $0.\ov 67$. Let y = $0.\ov 67$. Multiply by 100, 100y = 67 + $0.\ov 67$ which is same as 67 + y. So 99y = 67, ⇒ y = $67/99$. So answer = ${67/99}$.

Let the numbers be $a$ and $14 - a$. We are given their product as a × (14 - a) = 45, which is a quadratic expression that can be simplified to $(a - 9) × (5 - a) = 0$. So the numbers could be 9 and 5.

The first term is 81, common difference is d = -9, n-th term is 0. So $0 = 81 + (n - 1) × -9$ which gives $n = 1 + {81/9} = 10$.

Let the number be ab. When it is reversed and added to itself we get (10a + b) + (10b + a) = 11 × (a + b). We are given 55 = 11 × (a + b) ⇒ $a + b = 55 / 11 = 5$, so the digits are $a$ and $5 - a$. We are given their product as a × (5 - a) = 6, which is a quadratic expression that can be simplified to $(a - 3) × (2 - a) = 0$. So the number could be 32 or 23.

Let the numbers be 2n - 2, 2n and 2n + 2. The sum is 6n = 3 x 2n = 3 x middle. We are given 3 x middle = 672, ⇒ middle = $672/3$, i.e., middle = 224.