**ICSE Solutions for Class 10 Mathematics – Ratio and Proportion**

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**Formulae**

- When two or more ratios are multiplied together, they are said to be compounded.

- A ratio compounded with itself is called duplicate ratio of the given ratio.

- The reciprocal ratio a : b is b : a.
**Proportion:**An equality of two ratios is called a proportion.

- The quantities a, b, c and d are called the terms of the proportion; a, b, c and d are the first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).

⇒ product of extreme terms = product of middle terms.

Thus, if the four quantities are in proportion then the product of extreme terms = product of middle terms. This is called cross product rule. **Fourth proportional:**If a, b, c and d are in proportion then d is called the fourth proportional.

- In particular, three (non-zero) quantities of the same kind, a, b and c are said to be in continued proportion iff the ratio of a to b is equal to the ratio of b to c.

**First proportional:**If a, b are c are in continued proportion, then a is called the first proportional.

**Third proportional:**If a, b and c are in continued proportion, then c is called the third proportional.

**Mean proportional:**If a, b and c are in continued proportion, then b is called the mean proportional of a and c.

Thus, if b is the mean proportional of a and c, then

Hence, the mean proportion between two numbers is the positive square root of their product.

**Properties of Ratio & Proportion:**

**Determine the Following**

**Question 1.** Which is greater 4 : 5 or 19 : 25.

**Question 2.** Arrange 5 : 6, 8 : 9, 13 : 18 and 7 : 12 in ascending order of magnitude.

**Question 3.** Find:

**Question 4.** If 3x – 2y – 7z = 0 and 2x + 3y – 5z = 0 find x : y : z.

**Question 5.** Two numbers are in the ratio of 3 : 5. If 8 is added to each number, the ratio becomes 2 : 3. Find the numbers.

**Question 6.** If x : y = 2 : 3, find the value of (3x + 2y) : (2x + 5y).

**Question 7.** Divide Rs. 720 between Sunil, Sbhil and Akhil. So that Sunil gets 4/5 of Sohil’s and Akhil’s share together and Sohil gets 2/3 of Akhil’s share.

**Question 8.** The ratio between two numbers is 3 : 4. If their L.C.M., is 180. Find the numbers.

**Question 10.** Find the compound ratio of the following:

(i) If A : B = 4 : 5, B : C = 6 : 7 and C : D = 14 : 15. Find A : D.

(ii) If P : Q = 6 : 7, Q : R = 8 : 9 find P : Q : R.

**Question 11.** Find:

(i) The duplicate ratio of 7 : 9

(ii) The triplicate ratio of 3 : 7

(iii) The sub-duplicate ratio of 256 : 625

(iv) The sub-triplicate ratio of 216 : 343

(v) The reciprocal ratio of 8 : 15.

**Question 13.** (i) What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional ?

(ii) What least number must be added to each of the numbers 5, 11, 19 and 37, so that they are in proportion ?

**Question 15.** The work done by (x – 3) men in (2x + 1) days and the work done by (2x + 1) men in (x + 4) days are in the ratio of 3 : 10. Find the value of x.

**Question 16.** Find the third proportional to:

**Question 17.** Find the fourth proportional to:

**Question 18.** Find the two numbers such that their mean proprtional is 24 and the third proportinal is 1,536.

**Question 19.** Using componendo and idendo, find the value of x

**Question 20.** Solve for x:

**Question 21.** Using the properties of proportion, solve for x, given.

**Question 25.** Find the value of

**Prove the Following **

**Question 1.** If (a – x) : (b – x) be the duplicate ratio of a : b show that:

**Question 2.** If a : b with a ≠ b is the duplicate ratio of a + c : b + c, show that c^{2} = ab.

**Question 3.** If a : b = 5 : 3, show that (5a + 8b) : (6a – 7b) = 49 : 9.

**Question 7.** If x and y be unequal and x : y is the duplicate ratio of (x + z) and (y + z) prove that z is mean proportional between x and y.

**Question 8.** If ax = by = cz, prove that

**Question 15.** If a, b, c, d are in continued proportion, prove that

**Question 16.** If q is the mean proportional between p and r, prove that

**Question 21.** If a, b, c, d are in continued proportion, prove that:

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