Problems Based on Numbers - Arithmetic

Question to obtain the formula: When t times of a number is decreased by d, it gives y, then find the number.

[Here 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield]

Solution

Let the number = n

Thus, according to question,

t × n –

⇒

∴ n = ^{y + d}⁄_t

Thus, in such a situation, the required number can be found by using the above given formula

Question (1) If thrice of a number is decreased by 8, it gives 82, then find the number.

Answer

Given,

3 × Number – 8 = 82

Then number = ?

Let the number be a

Thus, as per question,

3 × a – 8 = 82

⇒ 3 a = 82 + 8

⇒ 3 a = 90

∴ a = ⁹⁰⁄₃

⇒ a = 30

Thus, required number = 30 Answer

Alternate Method

This question can be solved using the formula, Number = ^{y + d}⁄_t

Where 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield

Here in the question,

t = 3, d = 8, y = 82, thus number = ?

By using the formula Number = ^{y + d}⁄_t

⇒ Number = ^{82 + 8}⁄₃

⇒ Number = ⁹⁰⁄₃

⇒ Number = 30

Thus, required number = 30 Answer

Question (2) When four times of a number is decreased by 15, it gives 85. Then, what will be the number?

Answer

Given,

4 × Number – 15 = 85

Then number = ?

Let the number be a

Thus, as per question,

4 × a – 15 = 85

⇒ 4 a = 85 + 15

⇒ 4 a = 100

∴ a = ¹⁰⁰⁄₄

⇒ a = 25

Thus, required number = 25 Answer

Alternate Method

This question can be solved using the formula, Number = ^{y + d}⁄_t

Where 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield

Here in the question,

t = 4, d = 15, y = 85, thus number = ?

By using the formula Number = ^{y + d}⁄_t

⇒ Number = ^{85 + 15}⁄₄

⇒ Number = ¹⁰⁰⁄₄

⇒ Number = 25

Thus, required number = 25 Answer

Question (3) When 5 times of a number is decreased by 20, it gives 130. Find the number.

Answer

Given,

5 × Number – 20 = 130

Then number = ?

Let the number be a

Thus, as per question,

5 × a – 20 = 130

⇒ 5 a = 130 + 20

⇒ 5 a = 150

∴ a = ¹⁵⁰⁄₅

⇒ a = 30

Thus, required number = 30 Answer

Alternate Method

This question can be solved using the formula, Number = ^{y + d}⁄_t

Where 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield

Here in the question,

t = 5, d = 20, y = 130, thus number = ?

By using the formula Number = ^{y + d}⁄_t

⇒ Number = ^{130 + 20}⁄₅

⇒ Number = ¹⁵⁰⁄₅

⇒ Number = 30

Thus, required number = 30 Answer

Question (4) When 15 times of a number is decreased by 120, it gives 2130. Find the number.

Answer

Given,

15 × Number – 120 = 2130

Then number = ?

Let the number be a

Thus, as per question,

15 × a – 120 = 2130

⇒ 15 a = 2130 + 120

⇒ 15 a = 2250

∴ a = ²²⁵⁰⁄₁₅

⇒ a = 150

Thus, required number = 150 Answer

Alternate Method

This question can be solved using the formula, Number = ^{y + d}⁄_t

Where 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield

Here in the question,

t = 15, d = 120, y = 2130, thus number = ?

By using the formula Number = ^{y + d}⁄_t

⇒ Number = ^{2130 + 120}⁄₁₅

⇒ Number = ²²⁵⁰⁄₁₅

⇒ Number = 150

Thus, required number = 150 Answer

Question (5) When 20 times of a number is decreased by 135, it gives 1065. Find the number.

Answer

Given,

20 × Number – 135 = 1065

Then number = ?

Let the number be a

Thus, as per question,

20 × a – 135 = 1065

⇒ 20 a = 1065 + 135

⇒ 20 a = 1100

∴ a = ¹¹⁰⁰⁄₂₀

⇒ a = 55

Thus, required number = 55 Answer

Alternate Method

This question can be solved using the formula, Number = ^{y + d}⁄_t

Where 't' denotes the times, 'd' denotes the decreased, and 'y' denotes the yield

Here in the question,

t = 20, d = 135, y = 1065, thus number = ?

By using the formula Number = ^{y + d}⁄_t

⇒ Number = ^{1065 + 135}⁄₂₀

⇒ Number = ¹¹⁰⁰⁄₂₀

⇒ Number = 55

Thus, required number = 55 Answer