ICSE Class 8 Maths Algebra Chapter 02 Formulae

Formulae

A formula is a relation between certain quantities, expressed with the help of variables and mathematical symbols.

Examples

(i) "When a number is multiplied by 7 and 20 is deducted from the product, the result is 5 more than twice the number." If we denote the number by x, we can represent this statement by the formula, \(7x - 20 = 2x + 5\).

(ii) The volume (V) of a cuboid is the product of its length (l), breadth (b) and height (h). This can be expressed by the formula \(V = lbh\).

(iii) The area (A) of a rectangle is the product of its length (l) and its breadth (b). This can be expressed as \(A = l \times b\).

Framing A Formula

Steps

1. Use variables such as \(a, b, x, y, A, R\) and \(T\) for the quantities for which you want to frame the formula. Certain symbols are traditionally used to denote certain variables.

Examples

(i) V is used for volume, A for area, l for length, b for breadth, h for height and S for surface area in mensuration.

(ii) \(v\) and \(u\) are used for speed, s for distance, and t for time in physics.

(iii) P is used for principal, A for amount, R for rate, I for interest and T for time in arithmetic.

2. Use the rules or conditions relevant to the context to establish a relationship between the variables.

Solved Examples

Example 1

Ram is 22 years older than his son Rohit. After 8 years his age will be 5 years more than twice the age of Rohit. Frame a formula for these statements.

Solution

Let Rohit's present age = x years.

Then Ram's age = (x + 22) years.

After 8 years, Rohit's age = (x + 8) years

and Ram's age = [(x + 22) + 8] years = (x + 30) years.

From the question, \(x + 30 = 2(x + 8) + 5\).

This is the required formula.

Teacher's Note

Understanding age relationships helps us in solving real-world problems, such as when parents and children compare their ages or when we need to calculate someone's age after a certain number of years.

Example 2

In a two-digit number, the digit in the tens place exceeds the digit in the units place by 5. Write a formula for the number.

Solution

Let the digit in the units place be x.

Then the digit in the tens place = x + 5.

\[\therefore \text{the number} = 10 \times \text{digit in tens place} + \text{digit in units place}\]

\[= 10(x + 5) + x = 11x + 50.\]

Teacher's Note

Working with two-digit numbers and their digit relationships is fundamental in understanding place value, which is used in daily transactions and when reading numbers on price tags or addresses.

Remember These

1. A formula is a relation of equality or inequality between two or more quantities (or variables).

2. To frame a formula for a statement, we use literals (or variables) to represent the quantities concerned and express the relation between the quantities by an equality (or inequality).

Exercise 2A

1. Frame A Formula For Each Of The Following Statements.

(i) If 4 is subtracted from twice a certain integer n, the result is greater than the integer by 6.

(ii) In a two-digit number the digit in the tens place is 2 more than the digit in the units place. The number is seven times the sum of its digits.

(iii) The sum of three consecutive even numbers equals four times the smallest of the numbers (n).

(iv) A man's age is 20 years more than that of his son and the sum of their ages is 80 years.

(v) Sam is x years old. In 3 years, he will be thrice as old as he is now.

(vi) Annie is x years old. Aru is twice as old as her and Kris is 3 years younger than her. The sum of their ages is 20 years.

2. Express The Following As Formulae.

(i) The area A of a triangle is half of the product of its base b and altitude h.

(ii) The larger of two supplementary angles measures 30- more than the smaller angle, which measures x-.

(iii) The length of a rectangle is 2 m less than three times its width and its perimeter is six times its width.

3. Sachin has an average score of 62 runs in x innings and an average of 58 runs in y innings. Find his average score A for x and y innings.

4. Ravi earns Rs x per day on weekdays and Rs 50 more than the normal rate when he works on Sundays. Frame a formula for his earnings E in a 30-day month of 4 Sundays when he works on 2 Sundays in addition to weekdays.

5. Ramesh earns a profit of Rs 200 by selling 16 toys at the rate of Rs x per toy. If the cost price of each toy is Rs 50, frame a formula for the profit.

6. Samir bought a pencil and a sharpener for Rs 10. The pencil cost Rs 2 less than half the price of the sharpener. Frame a formulae to express this, taking the cost of the sharpener to be Rs x.

Teacher's Note

Formulas help us convert word problems into mathematical statements, which is essential in budgeting, calculating earnings, and understanding pricing in everyday shopping situations.

7. In all, 200 tickets were sold for a charity show. Adult tickets cost Rs 50 each and student tickets cost Rs 20 each. If the number of adult tickets sold was x, construct a formula for the income I (in rupees) from the show.

8. A man weighing 85 kg steps on a weighing scale while carrying a briefcase. If the scale reads x kg, frame a formula for the weight W (in kg) of the briefcase.

Answers

1. (i) 2n - 4 = n + 6 (ii) 10(a + 2) + a = 7[(a + 2) + a] (iii) n + (n + 2) + (n + 4) = 4n (iv) (x + 20) + x = 80

(v) x + 3 = 3x (vi) 2x + x + (x - 3) = 20

2. (i) \(A = \frac{1}{2} bh\) (ii) \((x + 30-) + x = 180-\) (iii) \(2[(3x - 2) + x] = 6x\)

3. \(A = \frac{62x + 58y}{x + y}\) runs

4. \(E = \text{Rs } [26x + 2(x + 50)]\)

5. \(16x - 50 \times 16 = 200\)

6. \(\left(\frac{x}{2} - 2\right) + x = 10\)

7. \(I = 50x + 20(200 - x)\)

8. \(W = x - 85\)

Subject Of A Formula

When one quantity (or variable) is expressed in terms of other quantities (or variables), the quantity (or variable) thus expressed is called the subject of the formula.

Examples

(i) The area (A) of a square of side a is \(A = a^2\).

Here, A is the subject of the formula.

(ii) If P, l and b denote the perimeter, length and breadth of a rectangle respectively then \(P = 2(l + b)\).

Here, P is the subject of the formula.

Changing The Subject Of A Formula

In the formula \(V = l \times b \times h\) for the volume of a cuboid, the subject is V. To express l in terms of the volume V, breadth b and height h, we write the formula as

\[l = \frac{V}{b \times h}.\]

The subject of the formula is now l.

This process of transforming a formula is called changing the subject of the formula.

Example

Change the subject of the formula \(v = u + ft\) to (i) f and (ii) t.

Solution

(i) Given, \(v = u + ft\).

\[\therefore v - u = ft \text{ or } f = \frac{v - u}{t}.\]

Now, f is the subject of the formula.

(ii) Again, \(ft = v - u\) or \(t = \frac{v - u}{f}\).

Here, t is the subject of the formula.

Teacher's Note

Changing the subject of formulas is used in physics when converting between different equations, such as calculating velocity, acceleration, or distance in motion problems.

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