ICSE Class 7 Maths Chapter 17 Formula Change of Subject of a Formula

Chapter 17

Formula: Change Of Subject Of A Formula

(Including Substitution)

17.1 Formula

A formula is an equation, which shows the relationship between two or more quantities (variables).

Infact, a formula is a translation from words to symbols.

For example:

1. Formula showing the relationship between the area of rectangle (A), its length (l) and its breadth (b) is:

\[A = l \times b\]

2. Formula showing the relationship between the distance (d) covered by a body in time (t) and with velocity (v) is:

\[d = v \times t\]

Example 1

Frame a formula for each of the following:

(i) Seven more than a certain number is twenty.

(ii) After seven years, the age of father will be three times the age of his son.

Solution

(i) Let the number be x.

Then the formula for the given statement is:

\[x + 7 = 20\]

(Ans.)

(ii) Let the present age of father = x years and the present age of son = y years.

After 7 years:

Father's age will be x + 7 years and son's age will be y + 7 years.

Formula for the given statement is:

\[x + 7 = 3(y + 7)\]

(Ans.)

Example 2

On monday, Rohit worked at a shop for 11 hours, out of which 8 hours was normal work and remaining hours was overtime. If he gets ₹ x for each hour of normal work and ₹ 2x for each hour of overtime, find (in terms of x) the total amount of money he gets on monday.

Solution

Number of hours of normal work = 8 hours

and, for each hour of normal work, he gets ₹ x

Money, he gets for 8 hours of normal work = 8 × ₹ x = ₹ 8x

Number of hours of overtime = (11 - 8) hours = 3 hours

and, for each hour of overtime, he gets ₹ 2x

Money, he gets for 3 hours of overtime = 3 × ₹ 2x = ₹ 6x

The total money he gets on monday = ₹ 8x + ₹ 6x = ₹ 14x

(Ans.)

Teacher's Note

Formulas help us translate real-world situations like calculating earnings or measuring areas into mathematical expressions that can be reused and understood universally.

Exercise 17(A)

Frame a formula for each of the following statements:

1. D is the number of days in w weeks and p days.

2. Twelve less than thrice a certain number is twenty-four.

3. Half of a number added to \(\frac{1}{3}\) of the same number is 10.

4. When two is subtracted from twice of a certain number, the result is twenty-two.

5. If five is subtracted from a certain number and the difference is divided by fifteen, the result is three.

6. If a number is multiplied by nine and then two is subtracted from it, the result is 88.

7. The sum of three consecutive integers is seventy-eight.

8. The sum of three consecutive odd integers is fifty-seven.

9. Ajay went to a market with ₹ 500. He buys a tennis ball for ₹ 10 and spends ₹ 75 on a racket plus ₹ 5 on conveyance. He still has ₹ x left.

10. A worker is paid ₹ 3 per hour for normal work and double this rate for overtime. Form a formula to find his earnings in a week (6 days) of 8 hours per day of normal work plus total overtime during this week being ten hours.

11. The final velocity (v) of a body is the sum of its initial velocity (u) and the product of acceleration produced (a) and time (t).

12. A taxi, in Delhi, charges ₹ 23 for the first kilometre and then ₹ 12 per kilometre for the remaining distance. Form an equation, if the taxi-driver charges ₹ 203 for a distance of x kilometre.

13. Eight years hence, Geeta will be twice as old as her age 5 years ago. Taking Geeta's present age as x years, form an equation in terms of x.

14. Mr. Verma is an officer in a Central Government office, which works for 5 days in a week. Mrs. Verma is also an officer in a State Government office which works for 6 days in a week. If per day earning of Mr. Verma is ₹ 280 more than that of Mrs. Verma; form an equation to find one week's earnings of Mrs. and Mr. Verma. Assume that one day earning of Mr. Verma is ₹ x.

Teacher's Note

Framing formulas from word problems develops critical thinking and helps us model everyday situations mathematically.

17.2 Changing The Subject Of A Formula

The subject of a formula is the variable which is expressed in terms of other variables.

For example:

1. In formula \(A = l \times b\), A is expressed in terms of l and b, so, A is the subject of the formula.

2. In \(I = \frac{P \times R \times T}{100}\), I is expressed in terms of P, R and T, so, I is the subject of the formula.

To change the subject of a given formula means to obtain a formula for a particular (required) quantity.

For example:

1. The formula \(A = l \times b\) can be re-written as:

(i) \(l = \frac{A}{b}\); here l is the subject of the formula.

(ii) \(b = \frac{A}{l}\); here b is the subject of the formula.

2. The formula \(I = \frac{P \times R \times T}{100}\) can be re-written as:

(i) \(P = \frac{I \times 100}{R \times T}\); here P is the subject of the formula.

(ii) \(R = \frac{I \times 100}{P \times T}\); here R is the subject of the formula and so on.

For changing the subject of a given formula, we use the same steps as are used in solving the equations.

Example 3

Given: \(p = 2l + 2b\). Make b the subject.

Solution

\(p = 2l + 2b\)

\(\Rightarrow p - 2l = 2l + 2b - 2l\)

[Subtracting 2l from both the sides]

\(\Rightarrow p - 2l = 2b\)

\(\Rightarrow \frac{p - 2l}{2} = \frac{2b}{2}\)

[Dividing each side by 2]

\(\Rightarrow \frac{p - 2l}{2} = b\)

or, \(b = \frac{p - 2l}{2}\)

(Ans.)

Example 4

Given: \(a = \frac{b + c}{m}\). Make c the subject.

Solution

\(a = \frac{b + c}{m}\)

\(\Rightarrow am = b + c\)

or \(\Rightarrow b + c = am\)

\(\Rightarrow c = am - b\)

(Ans.)

Teacher's Note

Changing the subject of a formula is like rearranging a recipe - sometimes we want to find the flour amount instead of the total weight, which requires algebraic manipulation.

Exercise 17(B)

Change the subject for the following formulae to the indicated letter (variable):

1. \(x + 2y = m\); for y

2. \(v^2 = u^2 + 2as\); for s

3. \(A = \frac{1}{2}(a + b)h\); for h

4. \(s = \frac{n}{2}(a + l)\); for l

5. \(C = \frac{5}{9}(F - 32)\); for F

6. \(F = \frac{9}{5}C + 32\); for C

7. \(A = p(a + rt)\); for p

8. \(s = ut + \frac{1}{2}at^2\); a

9. \(s = \frac{n}{2}{2a + (n - 1)d}\); d

10. \(a = \frac{x - y}{x + y}\); x

11. \(\frac{m - a}{m + b} = \frac{2c}{3d}\); m

12. \(\frac{5x + 8y}{3y - x} = 2a\); y

17.3 To Evaluate The Unknown, Using Change Of Subject Of Formula And Substitution Methods

Steps: 1. Change, if required, the formula to the required subject.

2. In the new formula, substitute the values of the given quantities and simplify.

Example 5

Given: \(\frac{m + c}{m} = x\), find c, if x = 5 and m = 10.

Solution

Step 1: \(\frac{m + c}{m} = x \Rightarrow m + c = mx\)

\(\Rightarrow c = mx - m\)

Since, we are not asked to forme a formula for c, it can be done directly. So,

\(\frac{m + c}{m} = x \Rightarrow \frac{10 + c}{10} = 5\)

\(\Rightarrow 10 + c = 50\)

Step 2: Substituting \(x = 5\) and \(m = 10\);

we get: \(c = 10 \times 5 - 10 = 40\)

(Ans.)

\(\Rightarrow c = 50 - 10 = 40\) (Ans.)

Teacher's Note

Using formulas with substitution is how engineers and scientists solve real problems - like calculating the speed of a car or the heat required to boil water.

Exercise 17(C)

1. \(C = \frac{5}{9}(F - 32)\). Find F, if C = 40.

2. \(V = \frac{1}{3}\pi r^2 h\). Find h, if V = 110 cm3 and r = 4 cm.

3. \(A = \frac{1}{2}(l + b)h\). Find b, if A = 60, l = 6 and h = 10.

4. \(t = 4\sqrt{\frac{h}{32}}\); express h in terms of t. Then calculate h, if t = 12.

5. In the formula \(p = \pi r + 2r\), make r the subject. Hence, find r, if p = 40 and \(\pi\) = 3.142.

6. If \(2y = \frac{x + 3}{x - 1}\) and y = 3. Find x.

7. \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). Find v when u = 15 and f = 5.

8. If x = 3 and y = -1; find z, if:

(i) \(z = (x + y)^2 - 5(x - y)\)

(ii) \(z = 8xy + x^2 - y^2\)

(iii) \(z = x^3 - y^3 - 3x^2y + 3xy^2\)

9. Given: \(A = 2\pi r(r + h)\), find h, if A = 2816 cm2, \(\pi = 3\frac{1}{7}\) and r = 14 cm.

10. Given: a = 5, b = -3 and c = 2. Find m, if:

(i) \(m = abc + a^2 - b^2 + c^2\)

(ii) \(m = a^3 + b^3 + c^3 - 5ab - 6bc\)

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