ICSE Class 9 Maths Chapter 06 Framing of Formula

Framing Of Formula

Points To Remember

1. Formula - It is a relation between two or more variables.

2. Framing of Formula - We can frame a formula or relation between the given variables based on certain given conditions.

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

Exercise 6 (A)

Frame a formula for each of the following statements:

Q. 1. The area (A) of a rectangle is equal to the product of its length (l) and breadth (b)

Ans. \(A = l.b.\)

Q. 2. The circumference (C) of a circle is equal to \(\pi\) times its diameter (d).

Ans. \(C = \pi d.\)

Q. 3. The area (A) of a circular ring is \(\pi\) times the difference between the squares of outer radius (R) and inner radius (r).

Ans. \(A = \pi (R^2 - r^2)\)

Q. 4. The volume (V) of a cylinder is the product of \(\pi\), the square of the base radius (r) and the height (h).

Ans. \(V = \pi r^2 h.\)

Q. 5. The total surface area (S) of a cone is equal to the product of \(\pi\), the base radius (r) and the sum of the base radius (r) and slant height (l)

Ans. \(S = \pi r (r + l)\)

Q. 6. Nine-fifths of the temperature (C) in centigrade of a body increased by 32 is equal to its temperature in Fahrenheit (F).

Ans. \(F = \frac{9}{5} C + 32\)

Q. 7. The reciprocal of focal length (f) is equal to the sum of the reciprocals of the object distance (u) and the image distance (v).

Ans. \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\)

Q. 8. The number of diagonals (d) that can be drawn from one vertex of an n-sided polygon to all over vertices is equal to 3 less than the number of sides of the polygon.

Ans. \(d = (n - 3)\)

Q. 9. The distance (s metres) when a freely falling body covers in time (t seconds) is 4.8 times the square of time (t).

Ans. \(s = 4.8 t^2\)

Q. 10. The arithmetic mean (M) of three quantities a, b, c is equal to their sum divided by the number of quantities.

Ans. \(M = \frac{a + b + c}{3}\)

(Note: Here number of quantities is 3)

Q. 11. Frame the formula for finding the number of minutes (M) in x hours, y minutes and z seconds.

Ans. \(M = 60x + y + \frac{z}{60}\)

(Note: 1 hour = 60 minutes and 1 minute = 60 seconds)

Teacher's Note

Formulas help us convert real-world descriptions into mathematical relationships, much like translating a recipe's instructions into precise measurements for cooking.

Q. 12. A shopkeeper marks each article at Rs. m and gives 10% discount on the marked price. If the cost price of each article be Rs. c, find the formula for profit (P).

Sol. Marked price = Rs. m

Discount = 10%

Cost price = Rs. c

Gain = P

Gain = S.P. - C.P.

\(= \frac{MP \times (100 - \text{Discount} \%)}{100} - CP\)

\(= \frac{m \times 90}{100} - c\)

\(\therefore P = \frac{9}{10} m - c\) Ans.

Q. 13. A man buys n articles at a total cost of Rs. C and sells them at x paise each. Obtain the formula for gain percent.

Sol. Cost price of n articles = Rs. C

Selling price of 1 article = x paise

\(\therefore\) S.P. of n article = Rs. \(\frac{nx}{100}\)

\(\therefore\) Gain = S.P. - C.P.

\(= \frac{nx}{100} - C = \frac{nx - 100C}{100}\)

Gain \(\% = \frac{nx - 100C}{100} \times \frac{100}{C}\)

\(= \left(\frac{nx - 100C}{C}\right)\% \text{ Ans.}\)

Q. 14. A cricketer has an average score of 68 runs per inning in x innings and average of 53 runs per inning in y innings. Find the formula for the average score (A) per inning for all the innings.

Sol. Average score in x innings = 68 runs

\(\therefore\) Total runs in x innings = 68x runs

Similarly total runs in y innings at the rate of 53 runs per inning = 53y

Now total number of innings = x + y

and total no. of runs scored = 68x + 53y

\(\therefore\) Average (A) = \(\frac{68x + 53y}{x + y}\) Ans.

Q. 15. The hiring charges (h) for a taxi are Rs. 200 plus Rs. 12 per km for distances covered beyond 30 km. If the distance travelled be x km (x > 30) write the formula for the hiring charges.

Sol. Hiring charges = Rs. 200 + Rs. 12 per km. after 30 km.

Total distance travelled = x km (x > 30)

\(\therefore\) Hiring charges = Rs. 200

+ (x - 30) × 12

\(\therefore h = \text{Rs. } 200 + 12 (x - 30)\) Ans.

Q. 16. A telephone subscriber has to pay Rs. 450 as quarterly rent in addition to y paise for each call exceeding 125 calls. If he makes n calls in a quarter (n > 125), write down the formula for the total bill (M) in rupees.

Sol. Quarterly rent = Rs. 450

Rate of per call exceeding 125 calls = y paise

No. of calls in a quarter = n

\(\therefore\) Total bill (M) = Rs. 450

\(+ \frac{y}{100} (x - 125)\)

\(= \text{Rs. } 450 + \frac{y(n - 125)}{100}\) Ans.

Q. 17. The average weight of a set of p articles is a gm and the total weight of another set of q articles is b kg. Frame the formula for the average weight (A) of (p + q) articles in kg.

Sol. Total weight of p articles with an average of a gm. per article = \(\frac{ap}{1000}\) kg.

Teacher's Note

Just as a tailor measures fabric to estimate how much cloth is needed for a garment, formulas help us calculate quantities in business situations like shipping costs and utility bills.

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