Framing Algebraic Expressions — Exercise 13(A)

Express each of the following statements in algebraic form:

(i) The sum of 8 and $x$ is equal to $y$.

(ii) $x$ decreased by 5 is equal to $y$.

(iii) 15 multiplied by $m$ gives $3n$.

(iv) Product of 8 and $y$ is equal to $3x$.

(v) 30 divided by $b$ is equal to $p$.

(vi) $z$ decreased by $3x$ is equal to $y$.

(vii) 12 times of $x$ is equal to $5z$.

(viii) $3z$ subtracted from 45 is equal to $y$.

(ix) $8x$ divided by $y$ is equal to $2z$.

(x) $7y$ decreased by $5x$ gives $8z$.

Answer

(i) $\bm{8 + x = y}$

(ii) $\bm{x - 5 = y}$

(iii) $\bm{15 \times m = 3n}$

(iv) $\bm{8 \times y = 3x}$

(v) $\bm{\dfrac{30}{b} = p}$

(vi) $\bm{z - 3x = y}$

(vii) $\bm{12 \times x = 5z}$

(viii) $\bm{45 - 3z = y}$

(ix) $\bm{\dfrac{8x}{y} = 2z}$

(x) $\bm{7y - 5x = 8z}$

Write in the form of an algebraic expression:

Perimeter ($P$) of a rectangle is two times the sum of its length ($l$) and its breadth ($b$).

Answer

By formula,

Perimeter of rectangle = 2 × (length + breadth)

Hence, $\bm{P = 2(l + b)}$.

Write in the form of an algebraic expression:

Perimeter ($P$) of a square is four times its side.

Answer

By formula,

Perimeter of square = 4 × side

Hence, $\bm{P = 4 \times \text{side}}$.

Write in the form of an algebraic expression:

Area of a square is square of its side.

Answer

By formula,

Area of square ($A$) = side × side = (side)²

Hence, $\bm{A = (\text{side})^2}$.

Write in the form of an algebraic expression:

Surface area of a cube is six times the square of its edge.

Answer

By formula,

Surface area of cube ($S$) = 6 × (edge)²

Hence, $\bm{S = 6(\text{edge})^2}$.

Express each of the following as an algebraic expression:

The sum of $x$ and $y$ minus $m$.

Answer

The sum of $x$ and $y$ is $(x + y)$, and subtracting $m$ from it gives :

Hence, $\bm{x + y - m}$.

Express each of the following as an algebraic expression:

The product of $x$ and $y$ divided by $m$.

Answer

The product of $x$ and $y$ is $xy$, and dividing it by $m$ gives :

Hence, $\bm{\dfrac{xy}{m}}$.

Express each of the following as an algebraic expression:

The subtraction of $5m$ from $3n$ and then adding $9p$ to it.

Answer

Subtracting $5m$ from $3n$ gives $(3n - 5m)$, and then adding $9p$ to it gives :

Hence, $\bm{3n - 5m + 9p}$.

Express each of the following as an algebraic expression:

The product of 12, $x, y$ and $z$ minus the product of 5, $m$ and $n$.

Answer

The product of $12, x, y$ and $z$ is $12xyz$, and the product of $5, m$ and $n$ is $5mn$. Subtracting the second product from the first gives :

Hence, $\bm{12xyz - 5mn}$.

Express each of the following as an algebraic expression:

Sum of $p$ and $2r - s$ minus sum of $a$ and $3n + 4x$.

Answer

The sum of $p$ and $2r - s$ is $(p + 2r - s)$, and the sum of $a$ and $3n + 4x$ is $(a + 3n + 4x)$.

Subtracting the second sum from the first gives $(p + 2r - s) - (a + 3n + 4x)$.

Hence, $\bm{(p + 2r - s) - (a + 3n + 4x)}$.

Construct a formula for the following:

Total wages (₹ $W$) of a man whose basic wage is (₹ $B$) for $t$ hours a week plus overtime at (₹ $R$) per hour, if he works a total of $T$ hours.

Answer

Basic wage for $t$ hours = ₹ B

Since he works a total of $T$ hours, overtime = $(T - t)$ hours

Wages for overtime = ₹ $R(T - t)$

∴ Total wages = Basic wage + Overtime wages

⇒ $W = B + R(T - t)$

Hence, required formula is $\bm{W = B + R(T - t)}$.

From the formula $B = 2a^2 – b^2$, calculate the value of B when $a = 3$ and $b = -1$.

Answer

Substituting $a = 3$ and $b = -1$ in $B = 2a^2 - b^2$, we get :

$\Rightarrow B = 2(3)^{2} - (-1)^{2} \\[1em] \Rightarrow B = 2 \times 9 - 1 \\[1em] \Rightarrow B = 18 - 1 \\[1em] \Rightarrow B = 17.$

Hence, $\bm{B}$ = 17.

The wages ₹ $W$ of a man earning ₹ $x$ per hour for $t$ hours are given by the formula $W = xt$. Find his wages for working 40 hours at a rate of ₹ 39.45 per hour.

Answer

Substituting $x = 39.45$ and $t = 40$ in $W = xt$, we get :

$\Rightarrow W = 39.45 \times 40 \\[1em] \Rightarrow W = 1578.$

Hence, his wages = ₹ 1578.

The temperature in Fahrenheit scale is represented by $F$ and the temperature in Celsius scale is represented by C. If $F = \dfrac{9}{5} \times C + 32°$, find F when $C = 40°$.

Answer

Substituting $C = 40°$ in $F = \dfrac{9}{5} \times C + 32°$, we get :

$\Rightarrow F = \dfrac{9}{5} \times 40° + 32° \\[1em] \Rightarrow F = 9 \times 8° + 32° \\[1em] \Rightarrow F = 72° + 32° \\[1em] \Rightarrow F = 104°.$

Hence, $\bm{F}$ = 104°.

Find the average (A) of four quantities $p, q, r$ and $s$.

If A = 6, $p = 3, q = 5$ and $r = 7$; find the value of $s$.

Answer

Average of four quantities = $\dfrac{\text{Sum of the quantities}}{\text{Number of quantities}}$

∴ Required formula is $A = \dfrac{p + q + r + s}{4}$

Substituting $A = 6, p = 3, q = 5$ and $r = 7$, we get :

$\Rightarrow 6 = \dfrac{3 + 5 + 7 + s}{4} \\[1em] \Rightarrow 6 \times 4 = 15 + s \\[1em] \Rightarrow 24 = 15 + s \\[1em] \Rightarrow s = 24 - 15 \\[1em] \Rightarrow s = 9.$

Hence, $\bm{A = \dfrac{p + q + r + s}{4}}$ and $\bm{s = 9}$.

Find $T$, if $T = 2a - b$, $a = 7$ and $b = 3$.

Answer

Substituting $a = 7$ and $b = 3$ in $T = 2a - b$, we get :

$\Rightarrow T = 2(7) - 3 \\[1em] \Rightarrow T = 14 - 3 \\[1em] \Rightarrow T = 11.$

Hence, $\bm{T}$ = 11.