Problems Based on Numbers

Math-home

Question (3) The sum to two numbers is 5 times of the first number. If the first number is 10, then find the second number.

Solution

Given,

First number = 10

And, sum of two numbers = 5 × first number

Thus, second number = ?

Let the second number = x

Now, according to question,

First number + Second number = 5 × first number

⇒ 10 + x = 5 × 10

⇒ 10 + x = 50

Now, after transposing 10 to the RHS

⇒ x = 50 – 10

⇒ x = 40

Thus, the second number = 40 Answer

Check for the Answer

First number + Second number = 5 × first number

⇒ 10 + 40 = 50

[∵ As calculated above, the second number = 40]

Thus, LHS = RHS. Hence answer is correct.

Alternate Method

Since, sum of two numbers = 5 × first number

And first number = 10

Here total = 5 times

Thus, second number = 5 time of first number – 1 time of first number

= (5 – 1) times of first number

⇒ Second number = 4 times of first number

⇒ Second number = 4 × 10 = 40

Thus, Answer = 40

Alternate Method

Given,

First number = 10

And, sum of two numbers = 5 × first number

= 5 × 10 = 50

Thus, second number = ?

Thus, second number = Sum of two numbers – First number

= 50 – 10 = 40

Thus, Answer = 40

Shortcut Method

When sum of the number is given as `n` times of a number,

Then second number = (n – 1 ) × given number

Here, as given, sum of two numbers = 5 × of second number

Here, n = 5

And second number = 10

Thus, the required (first) number = (5 – 1) × number

= 4 × 10 = 40

Thus, Answer = 40

Question (4) The sum to two numbers is 50 times of the first number. If the first number is 20, then find the second number.

Solution

Given,

First number = 20

And, sum of two numbers = 50 × first number

Thus, second number = ?

Let the second number = x

Now, according to question,

First number + Second number = 50 × first number

⇒ 20 + x = 50 × 20

⇒ 20 + x = 1000

Now, after transposing 20 to the RHS

⇒ x = 1000 – 20

⇒ x = 980

Thus, the second number = 980 Answer

Check for the Answer

First number + Second number = 50 × first number

⇒ 20 + 980 = 50 × 20

⇒ 20 + 980 = 1000

[∵ As calculated above, the second number = 980]

Thus, LHS = RHS. Hence answer is correct.

Alternate Method

Since, sum of two numbers = 50 × first number

And first number = 20

Here total = 50 times

Thus, second number = 50 time of first number – 1 time of first number

= (50 – 1) times of first number

⇒ Second number = 49 times of first number

⇒ Second number = 49 × 20 = 980

Thus, Answer = 980

Alternate Method

Given,

First number = 20

And, sum of two numbers = 50 × first number

= 50 × 20 = 1000

Thus, second number = ?

Thus, second number = Sum of two numbers – First number

= 1000 – 20 = 980

Thus, Answer = 980

Shortcut Method

When sum of the number is given as `n` times of a number,

Then second number = (n – 1 ) × given number

Here, as given, sum of two numbers = 50 × of second number

Here, n = 50

And second number = 20

Thus, the required (first) number = (50 – 1) × number

= 49 × 20 = 980

Thus, Answer = 980

Question (5) The sum to two numbers is 25 times of the first number. If the first number is 25, then find the second number.

Solution

Given,

First number = 25

And, sum of two numbers = 25 × first number

Thus, second number = ?

Let the second number = x

Now, according to question,

First number + Second number = 25 × first number

⇒ 25 + x = 25 × 25

⇒ 25 + x = 625

Now, after transposing 25 to the RHS

⇒ x = 625 – 25

⇒ x = 600

Thus, the second number = 600 Answer

Check for the Answer

First number + Second number = 25 × first number

⇒ 25 + 600 = 25 × 25

⇒ 25 + 600 = 625

[∵ As calculated above, the second number = 625]

Thus, LHS = RHS. Hence answer is correct.

Alternate Method

Since, sum of two numbers = 25 × first number

And first number = 25

Here total = 25 times

Thus, second number = 25 time of first number – 1 time of first number

= (25 – 1) times of first number

⇒ Second number = 24 times of first number

⇒ Second number = 24 × 25 = 600

Thus, Answer = 600

Alternate Method

Given,

First number = 25

And, sum of two numbers = 25 × first number

= 25 × 25 = 625

Thus, second number = ?

Thus, second number = Sum of two numbers – First number

= 600 – 25 = 600

Thus, Answer = 600

Shortcut Method

When sum of the number is given as `n` times of a number,

Then second number = (n – 1 ) × given number

Here, as given, sum of two numbers = 25 × of second number

Here, n = 25

And second number = 25

Thus, the required (first) number = (25 – 1) × number

= 24 × 25 = 600

Thus, Answer = 600

Question (6) The difference between two numbers is 5 times of the second number. If the second number is 25, then find the first number.

Solution

Given,

Difference between two numbers = 5 × second number

The second number = 25

Thus, first number = ?

Let the first number = x

Thus, according to question,

First number – Second number = 5 × second number

⇒ x – 25 = 5 × 25

⇒ x – 25 = 125

Now, after transposing –25 to RHS

[After transposing of –25 to RHS, it becomes +25]

⇒ x = 125 + 25

⇒ x = 150

Thus, the first number = 150 Answer

Alternate Method

Given,

The second number = 25

Difference between two numbers = 5 × second number

= 5 × 25 = 125

⇒ Difference between two numbers = 125

Thus, first number = ?

Since, difference between two numbers = 125

And one of the number = 25

Thus, another number = difference + one given number

= 125 + 25 = 150

Thus, required (first) number = 150 Answer

Alternate Method

As given in the question,

Difference of two numbers = 5 × second number

And second (given) number = 25

Thus, first number = 5 time of second number + 1 time of second number

= (5 + 1) times of second (given) number

⇒ First number = 6 times of second (given) number

⇒ First number = 6 × 25 = 150

Thus, Answer = 150

Shortcut Method

When difference between the number is given as `n` times of a number,

Then second number = (n + 1 ) × given number

Here, as given, sum of two numbers = 5 × of second number

Here, n = 5

And second number = 25

Thus, the required (first) number = (5 + 1) × number

= 6 × 25 = 150

Thus, Answer = 150

Question (7) The difference between two numbers is 9 times of the second number. If the second number is 20, then find the first number.

Solution

Given,

Difference between two numbers = 9 × second number

The second number = 20

Thus, first number = ?

Let the first number = x

Thus, according to question,

First number – Second number = 9 × second number

⇒ x – 20 = 9 × 20

⇒ x – 20 = 180

Now, after transposing –20 to RHS

[After transposing of – 20 to RHS, it becomes +20]

⇒ x = 180 + 20

⇒ x = 200

Thus, the first number = 200 Answer

Alternate Method

Given,

The second number = 20

Difference between two numbers = 9 × second number

= 9 × 20 = 180

⇒ Difference between two numbers = 180

Thus, first number = ?

Since, difference between two numbers = 180

And one of the number = 20

Thus, another number = difference + one given number

= 180 + 20 = 200

Thus, required (first) number = 200 Answer

Alternate Method

As given in the question,

Difference of two numbers = 9 × second number

And second (given) number = 20

Thus, first number = 9 time of second number + 1 time of second number

= (9 + 1) times of second (given) number

⇒ First number = 10 times of second (given) number

⇒ First number = 10 × 20 = 200

Thus, Answer = 200

Shortcut Method

When difference of the number is given as `n` times of a number,

Then second number = (n + 1 ) × given number

Here, as given, sum of two numbers = 9 × of second number

Here, n = 9

And second number = 20

Thus, the required (first) number = (9 + 1) × number

= 10 × 20 = 200

Thus, Answer = 200

Question (8) The difference between two numbers is 21 times of the second number. If the second number is 70, then find the first number.

Solution

Given,

Difference between two numbers = 21 × second number

The second number = 70

Thus, first number = ?

Let the first number = x

Thus, according to question,

First number – Second number = 21 × second number

⇒ x – 70 = 21 × 70

⇒ x – 70 = 1470

Now, after transposing –70 to RHS

[After transposing of – 70 to RHS, it becomes +70]

⇒ x = 1470 + 70

⇒ x = 1540

Thus, the first number = 1540 Answer

Alternate Method

Given,

The second number = 70

Difference between two numbers = 21 × second number

= 21 × 70 = 1470

⇒ Difference between two numbers = 1470

Thus, first number = ?

Since, difference between two numbers = 1470

And one of the number = 70

Thus, another number = difference + one given number

= 1470 + 70 = 1540

Thus, required (first) number = 1540 Answer

Alternate Method

As given in the question,

Difference of two numbers = 21 × second number

And second (given) number = 70

Thus, first number = 21 time of second number + 1 time of second number

= (21 + 1) times of second (given) number

⇒ First number = 22 times of second (given) number

⇒ First number = 22 × 70 = 1540

Thus, Answer = 1540

Shortcut Method

When difference of the number is given as `n` times of a number,

Then second number = (n + 1 ) × given number

Here, as given, sum of two numbers = 21 × of second number

Here, n = 21

And second number = 70

Thus, the required (first) number = (21 + 1) × number

= 22 × 70 = 1540

Thus, Answer = 1540

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