Framing Algebraic Expressions — Exercise 13(E)

A number increased by 17 becomes 54. Find the number.

Answer

Let the number be $x$.

Given,

The number increased by 17 becomes 54.

$\Rightarrow x + 17 = 54 \\[1em] \Rightarrow x = 54 - 17 \\[1em] \Rightarrow x = 37$

Hence, the number is 37.

A number decreased by 8 equals 26; find the number.

Answer

Let the number be $x$.

Given,

The number decreased by 8 equals 26.

$\Rightarrow x - 8 = 26 \\[1em] \Rightarrow x = 26 + 8 \\[1em] \Rightarrow x = 34$

Hence, the number is 34.

One-fourth of a number when added to two-seventh of it, gives 135. Find the number.

Answer

Let the number be $x$.

Given,

One-fourth of the number when added to two-seventh of it, gives 135.

$\Rightarrow \dfrac{1}{4}x + \dfrac{2}{7}x = 135 \\[1em] \Rightarrow \dfrac{7x + 8x}{28} = 135 \\[1em] \Rightarrow \dfrac{15x}{28} = 135 \\[1em] \Rightarrow 15x = 135 \times 28 \\[1em] \Rightarrow x = \dfrac{135 \times 28}{15} \\[1em] \Rightarrow x = 9 \times 28 = 252$

Hence, the number is 252.

Two-fifth of a number when subtracted from three-fourth of it, gives 56. Find the number.

Answer

Let the number be $x$.

Given,

Two-fifth of the number when subtracted from three-fourth of it, gives 56.

$\Rightarrow \dfrac{3}{4}x - \dfrac{2}{5}x = 56 \\[1em] \Rightarrow \dfrac{15x - 8x}{20} = 56 \\[1em] \Rightarrow \dfrac{7x}{20} = 56 \\[1em] \Rightarrow 7x = 56 \times 20 \\[1em] \Rightarrow x = \dfrac{56 \times 20}{7} \\[1em] \Rightarrow x = 8 \times 20 = 160$

Hence, the number is 160.

A number is increased by 12 and the new number obtained is multiplied by 5. If the resulting number is 95, find the original number.

Answer

Let the original number be $x$.

Given,

The number is increased by 12 and the new number obtained is multiplied by 5. The resulting number is 95.

$\Rightarrow (x + 12) \times 5 = 95 \\[1em] \Rightarrow x + 12 = \dfrac{95}{5} \\[1em] \Rightarrow x + 12 = 19 \\[1em] \Rightarrow x = 19 - 12 \\[1em] \Rightarrow x = 7$

Hence, the original number is 7.

A number is increased by 26 and the new number obtained is divided by 3. If the resulting number is 18, find the original number.

Answer

Let the original number be $x$.

Given,

The number is increased by 26 and the new number obtained is divided by 3. The resulting number is 18.

$\Rightarrow \dfrac{x + 26}{3} = 18 \\[1em] \Rightarrow x + 26 = 18 \times 3 \\[1em] \Rightarrow x + 26 = 54 \\[1em] \Rightarrow x = 54 - 26 \\[1em] \Rightarrow x = 28$

Hence, the original number is 28.

The age of a man is 27 years more than the age of his son. If the sum of their ages is 47 years, find the age of the son and his father.

Answer

Let the age of the son be $x$ years, then the age of the man = ($x$ + 27) years.

Given,

The age of a man is 27 years more than the age of his son. The sum of their ages is 47 years.

$\Rightarrow x + (x + 27) = 47 \\[1em] \Rightarrow 2x + 27 = 47 \\[1em] \Rightarrow 2x = 47 - 27 \\[1em] \Rightarrow 2x = 20 \\[1em] \Rightarrow x = 10$

∴ Age of the man = $x$ + 27 = 10 + 27 = 37 years

Hence, the son is 10 years old and his father is 37 years old.

The difference between the ages of Gopal and his father is 26 years. If the sum of their ages is 56 years, find the ages of Gopal and his father.

Answer

Let the age of Gopal be $x$ years, then the age of his father = ($x$ + 26) years.

Given,

The difference between the ages of Gopal and his father is 26 years. The sum of their ages is 56 years.

$\Rightarrow x + (x + 26) = 56 \\[1em] \Rightarrow 2x + 26 = 56 \\[1em] \Rightarrow 2x = 56 - 26 \\[1em] \Rightarrow 2x = 30 \\[1em] \Rightarrow x = 15$

∴ Age of his father = $x$ + 26 = 15 + 26 = 41 years

Hence, Gopal is 15 years old and his father is 41 years old.

When two consecutive natural numbers are added, the sum is 31; find the numbers.

Answer

Let the two consecutive natural numbers be $x$ and $x + 1$.

Given,

When two consecutive natural numbers are added, the sum is 31.

$\Rightarrow x + (x + 1) = 31 \\[1em] \Rightarrow 2x + 1 = 31 \\[1em] \Rightarrow 2x = 31 - 1 \\[1em] \Rightarrow 2x = 30 \\[1em] \Rightarrow x = 15$

∴ The numbers are 15 and 15 + 1 = 16.

Hence, the numbers are 15 and 16.

When three consecutive natural numbers are added, the sum is 66, find the numbers.

Answer

Let the three consecutive natural numbers be $x, x + 1$ and $x + 2$.

Given,

When three consecutive natural numbers are added, the sum is 66.

$\Rightarrow x + (x + 1) + (x + 2) = 66 \\[1em] \Rightarrow 3x + 3 = 66 \\[1em] \Rightarrow 3x = 66 - 3 \\[1em] \Rightarrow 3x = 63 \\[1em] \Rightarrow x = 21$

∴ The numbers are 21, 21 + 1 = 22 and 21 + 2 = 23.

Hence, the numbers are 21, 22 and 23.

One-fourth of a number added to one-sixth of it is 15. Find the number.

Answer

Let the number be $x$.

Given,

One-fourth of the number added to one-sixth of it is 15.

$\Rightarrow \dfrac{1}{4}x + \dfrac{1}{6}x = 15 \\[1em] \Rightarrow \dfrac{3x + 2x}{12} = 15 \\[1em] \Rightarrow \dfrac{5x}{12} = 15 \\[1em] \Rightarrow 5x = 15 \times 12 \\[1em] \Rightarrow x = \dfrac{15 \times 12}{5} \\[1em] \Rightarrow x = 3 \times 12 = 36$

Hence, the number is 36.

A whole number is increased by 7 and the number so obtained is multiplied by 5; the result is 45. Find the whole number.

Answer

Let the whole number be $x$.

Given,

The whole number is increased by 7 and the number so obtained is multiplied by 5, giving the result 45.

$\Rightarrow (x + 7) \times 5 = 45 \\[1em] \Rightarrow x + 7 = \dfrac{45}{5} \\[1em] \Rightarrow x + 7 = 9 \\[1em] \Rightarrow x = 9 - 7 \\[1em] \Rightarrow x = 2$

Hence, the whole number is 2.

The age of a man and the age of his daughter differ by 23 years, and the sum of their ages is 41 years. Find the age of the man.

Answer

Given,

The age of a man and the age of his daughter differ by 23 years.

Let the age of the daughter be $x$ years, then the age of the man = ($x$ + 23) years.

Given,

The sum of their ages is 41 years.

$\Rightarrow x + (x + 23) = 41 \\[1em] \Rightarrow 2x + 23 = 41 \\[1em] \Rightarrow 2x = 41 - 23 \\[1em] \Rightarrow 2x = 18 \\[1em] \Rightarrow x = 9$

∴ Age of the man = $x$ + 23 = 9 + 23 = 32 years

Hence, the age of the man is 32 years.

The difference between two numbers is 15. Taking the smaller number as $x$, find :

(i) the expression for the larger number.

(ii) the larger number if the sum of these numbers is 71.

Answer

Let the smaller number be $x$.

(i) Since, the difference between the two numbers is 15,

Larger number = smaller number + 15 = $x$ + 15

Hence, the expression for the larger number is ($x$ + 15).

(ii) Given, the sum of the two numbers is 71.

$\Rightarrow x + (x + 15) = 71 \\[1em] \Rightarrow 2x + 15 = 71 \\[1em] \Rightarrow 2x = 71 - 15 \\[1em] \Rightarrow 2x = 56 \\[1em] \Rightarrow x = 28$

∴ Larger number = $x$ + 15 = 28 + 15 = 43

Hence, the larger number is 43.

The difference between two numbers is 23. Taking the larger number as $x$, find :

(i) the expression for smaller number.

(ii) the smaller number, if the sum of these two numbers is 91.

Answer

Let the larger number be $x$.

Given,

The difference between two numbers is 23.

(i) Since the difference between the two numbers is 23,

Smaller number = larger number − 23 = $x$ − 23

Hence, the expression for the smaller number is ($x$ − 23).

(ii) Given, the sum of the two numbers is 91.

$\Rightarrow x + (x - 23) = 91 \\[1em] \Rightarrow 2x - 23 = 91 \\[1em] \Rightarrow 2x = 91 + 23 \\[1em] \Rightarrow 2x = 114 \\[1em] \Rightarrow x = 57$

∴ Smaller number = $x$ − 23 = 57 − 23 = 34

Hence, the smaller number is 34.

Find three consecutive integers whose sum is 78.

Answer

Given,

Sum of three consecutive integers is 78.

Let the three consecutive integers be $x, x + 1$ and $x + 2$.

$\Rightarrow x + (x + 1) + (x + 2) = 78 \\[1em] \Rightarrow 3x + 3 = 78 \\[1em] \Rightarrow 3x = 78 - 3 \\[1em] \Rightarrow 3x = 75 \\[1em] \Rightarrow x = 25$

∴ The integers are 25, 25 + 1 = 26 and 25 + 2 = 27.

Hence, the three consecutive integers are 25, 26 and 27.

The sum of three consecutive numbers is 54. Taking the middle number as $x$, find the three numbers.

Answer

Let the middle number be $x$, then the three consecutive numbers are $x - 1, x$ and $x + 1$.

Given,

The sum of three consecutive numbers is 54.

$\Rightarrow (x - 1) + x + (x + 1) = 54 \\[1em] \Rightarrow 3x = 54 \\[1em] \Rightarrow x = 18$

∴ The numbers are 18 − 1 = 17, 18 and 18 + 1 = 19.

Hence, the three numbers are 17, 18 and 19.