Generate algebraic expressions for the following:

(i) The product of a and b added to their sum.

(ii) The quotient of x by 8 is multiplied by y.

(iii) Thrice x added to y squared.

(iv) One-third of x multiplied by the sum of p and q.

(v) The number to be added to x + y to make it equal to p.

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

(vii) Five times m subtracted from the sum of p and thrice x.

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

(ix) The sum of 7 numbers each equal to p.

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

(i) The product of a and b added to their sum.

Product of a and b = ab

Sum of a and b = (a + b)

∴ The expression is (a + b) + ab

(ii) The quotient of x by 8 is multiplied by y.

Quotient of x by 8 = $\dfrac{x}{8}$

Multiplied by y = $\dfrac{x}{8} \times y$

∴ The expression is $\dfrac{x}{8} \times y$

(iii) Thrice x added to y squared.

Thrice x = 3x

y squared = y²

∴ The expression is y² + 3x

(iv) One-third of x multiplied by the sum of p and q.

One-third of x = $\dfrac{1}{3}x$

Sum of p and q = (p + q)

∴ The expression is $\dfrac{1}{3}x(p + q)$

(v) The number to be added to x + y to make it equal to p.

Let number = ?

x + y + ? = p

⇒ ? = p − (x + y)

∴ The expression is p - (x + y)

(vi) From a rod (p + q) units in length, n equal pieces are cut. Find the length of each piece.

Total length = p + q

Number of pieces = n

Length of each piece = $\dfrac{\text{Total length}}{\text{Number of pieces}}$

Length of each piece = $\dfrac{p + q}{n}$

∴ The expression is $\dfrac{p + q}{n}$

(vii) Five times m subtracted from the sum of p and thrice x.

Five times m = 5m

Sum of p and thrice x = (p + 3x)

∴ The expression is (p + 3x) - 5m

(viii) The number obtained when m times the difference of x and y is subtracted from n times the sum of x and y.

Difference of x and y = (x − y)

m times difference = m(x - y)

Sum of x and y = x + y

n times sum = n(x + y)

∴ The expression is n(x + y) - m(x - y)

(ix) The sum of 7 numbers each equal to p.

Adding p seven times is the same as 7 multiplied by p.

∴ The expression is 7p

(x) The product of three numbers a, b and c subtracted from the sum of x and y.

Product of a, b, c = abc

Sum of x and y = x + y

∴ The expression is (x + y) - abc