ICSE Solutions for Class 10 Mathematics – Ratio and Proportion
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- 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 c2 = 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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