Question

8. The tens digit of a number is 1 more than its ones place. If the sum of the number and the
number obtained by interchanging its digits is 33, find the number.
9. The numerator of a fraction is 3 less than its denominator. When both the numerator and the
denominator are increased by 1 the fraction becomes -. Find the fraction.
10. The present ages of Gagan and Albert are in the ratio 2:3. Four years from now their ages will
be in the ratio 5:7, then find their present ages.​

Answer 1

Answer:

8. $\boxed{\red{\sf\:Two\:-\:digit\:number\:=\:21}}$

9. $\boxed{\pink{\sf\:The\:required\:fraction\:is\:\frac{5}{8}}}$

10. $\boxed{\blue{\sf\:The\:present\:ages\:of\:Gagan\:and\:Albert\:are\:16\:years\:\&\:24\:years\:respectively}}$

Step-by-step-explanation:

8.

Let the digit at tens place be x.

And the digit at units place be y.

The two digit number = 10x + y.

The number obtained by interchanging the digits = 10y + x.

From the first condition,

$\sf\:x\:=\:y\:+\:1\:\:\:-\:-\:(\:1\:)$

From the second condition,

$\sf\:10x\:+\:y\:+\:10y\:+\:x\:=\:33\\\\:\implies\sf\:11x\:+\:11y\:=\:33\\\\:\implies\sf\:x\:+\:y\:=\:3\:\:\:-\:-\:[\:Dividing\:by\:11\:]\\\\:\implies\sf\:y\:+\:1\:+\:y\:=\:3\:\:\:-\:-\:-\:[\:From\:(\:1\:)\:]\\\\:\implies\sf\:2y\:+\:1\:=\:3\\\\:\implies\sf\:2y\:=\:3\:-\:1\\\\:\implies\sf\:2y\:=\:2\\\\:\implies\sf\:y\:=\:\cancel{\frac{2}{2}}\\\\:\implies\boxed{\red{\sf\:y\:=\:1}}$

By substituting y = 1 in equation ( 1 ), we get,

$\sf\:x\:=\:y\:+\:1\:\:\:-\:-\:(\:1\:)\\\\:\implies\sf\:x\:=\:1\:+\:1\\\\:\implies\boxed{\red{\sf\:x\:=\:2}}$

$\sf\:Two\:-\:digit\:number\:=\:10x\:+\:y\\\\:\implies\sf\:Two\:-\:digit\:number\:=\:10\:\times\:2\:+\:1\\\\:\implies\sf\:Two\:-\:digit\:number\:=\:20\:+\:1\\\\:\implies\boxed{\red{\sf\:Two\:-\:digit\:number\:=\:21}}$

$\rule{200}{1}$

9. Correct Question:

The numerator of a fraction is 3 less than its denominator. When both the numerator and the

denominator are increased by 1, the fraction becomes $\sf\dfrac{2}{3}$. Find the fraction.

Step-by-step-explanation:

Let the numerator of the fraction be x.

And the denominator of the fraction be y.

From the first condition,

$\sf\:x\:=\:y\:-\:3\:\:\:-\:-\:-\:(\:1\:)$

From the second condition,

$\sf\:\dfrac{x\:+\:1}{y\:+\:1}\:=\:\dfrac{2}{3}\\\\:\implies\sf\:3\:(\:x\:+\:1\:)\:=\:2\:(\:y\:+\:1\:)\\\\:\implies\sf\:3x\:+\:3\:=\:2y\:+\:2\\\\:\implies\sf\:3x\:-\:2y\:=\:2\:-\:3\\\\:\implies\sf\:3x\:-\:2y\:=\:-\:1\\\\:\implies\sf\:3\:(\:y\:-\:3\:)\:-\:2y\:=\:-\:1\:\:\:[\:From\:(\:1\:)\:]\\\\:\implies\sf\:3y\:-\:9\:-\:2y\:=\:-\:1\\\\:\implies\sf\:y\:-\:9\:=\:-\:1\\\\:\implies\sf\:y\:=\:-\:1\:+\:9\\\\:\implies\boxed{\pink{\sf\:y\:=\:8}}$

Now, by substituting y = 8 in equation ( 1 ), we get,

$\sf\:x\:=\:y\:-\:3\:\:\:-\:-\:(\:1\:)\\\\:\implies\sf\:x\:=\:8\:-\:3\\\\:\implies\boxed{\pink{\sf\:x\:=\:5}}$

$\boxed{\pink{\sf\:The\:required\:fraction\:is\:\frac{5}{8}}}$

$\rule{200}{1}$

10.

Let the present age of Gagan be x years.

And the present age of Albert be y years.

From the first condition,

$\sf\:x\::\:y\:=\:2\::\:3\\\\:\implies\sf\:\frac{x}{y}\:=\:\frac{2}{3}\\\\:\implies\sf\:3x\:=\:2y\\\\:\implies\sf\:x\:=\:\frac{2y}{3}\:\:\:-\:-\:(\:1\:)$

The age of Gagan after 4 years from now = ( x + 4 ) years.

And the age of Albert after 4 years from now = ( y + 4 ) years.

From the second condition,

$\sf\:\dfrac{x\:+\:4}{y\:+\:4}\:=\:\dfrac{5}{7}\\\\:\implies\sf\:7\:(\:x\:+\:4\:)\:=\:5\:(\:y\:+\:4\:)\\\\:\implies\sf\:7x\:+\:28\:=\:5y\:+\:20\\\\:\implies\sf\:7x\:-\:5y\:=\:20\:-\:28\\\\:\implies\sf\:7x\:-\:5y\:=\:-\:8\\\\:\implies\sf\:7\:(\:\frac{2y}{3}\:)\:-\:5y\:=\:-\:8\:\:\:-\:-\:-\:[\:From\:(\:1\:)\:]\\\\:\implies\sf\:\frac{14y}{3}\:-\:5y\:=\:-\:8\\\\:\implies\sf\:\dfrac{14y\:-\:15y}{3}\:=\:-\:8\\\\:\implies\sf\:-\:y\:=\:-\:8\:\times\:3\\\\:\implies\sf\:\cancel{-}\:y\:=\:\cancel{-}\:24\\\\:\implies\boxed{\blue{\sf\:y\:=\:24}}$

By substituting y = 24 in equation ( 1 ), we get,

$\sf\:x\:=\:\frac{2y}{3}\:\:\:-\:-\:-\:(\:1\:)\\\\\implies\sf\:x\:=\:\dfrac{2\:\times\:\cancel{24}}{\cancel3}\\\\:\implies\sf\:x\:=\:2\:\times\:8\\\\:\implies\boxed{\blue{\sf\:x\:=\:16}}$

$\boxed{\blue{\sf\:The\:present\:ages\:of\:Gagan\:and\:Albert\:are\:16\:years\:\&\:24\:years\:respectively}}$

Answer 2

the above answers is correct