Ratio & Proportion

1. TestBag TestBag Ratio & Proportion By: TestBag Faculty Grab the opportunity www.testbag.com

2. TestBag TestBag RATIO PROPORTION 1.1 Compound Ratio 1.2 Inverse Ratio 1.3 Comparison of Ratio 2.1 Forth Proportional 2.2 Third Proportional 2.3 Mean Proportional 2.4 Type of Proportional A.) Direct Proportion B.) Inverse Proportion www.testbag.com

3. TestBag TestBag • A ratio is an expression that compares quantities • relative to each other. What is ratio? • Can be written as x to y, x : y, or x/y. • The most common examples involve two quantities, • but any number of quantities can be compared.0 • Ratios are represented mathematically by separating • each quantity with a colon - www.testbag.com

4. TestBag TestBag The ratio of a to b is written as a : b or a/b = a÷b.Since a : b is a fraction, b can never be zero. The two quantities must be of the same kind. In a : b, a is antecedent and b is consequent. Example: The ratio 5 : 9 represents 5/9, here antecedent = 5, Consequent=9. The fraction a/b is usually different from the fraction b/a, So the order of the terms in a ratio is important. a b — ≠ — a b www.testbag.com

5. TestBag TestBag If the ratio between the first and the second quantitiesis a : b and the ratio between the second and third quantities is c : d, then the ratio among first, second and third quantities is given by ac : bc : bd.  Example: The sum of three numbers is 98. If the ratio between the first and second be 2 : 3 and that between the second and third be 5 : 8, then find the second number. Solution : The ratio among the three numbers is 2 : 3 5 : 8 10 : 15 : 24 ∴ The second number =30 98 = —————— ×15 10 + 15 + 24 www.testbag.com

6. TestBag TestBag If the ratio between then first and the second quantities is a : b the ratio between the second and the third quantities is c : d and the ratio between the third and the fourth quantities is e : f then the ratio among the first, second, third and fourth quantities is given by  b a 1st : 2nd = 2nd : 3rd = 3rd : 4th = d c e f 1st : 2nd : 3rd : 4th = ace : bce : bde : bdf www.testbag.com

7. TestBag TestBag Example : If A : B = 3 : 4, B : C = 8 : 10 and C : D = 15 : 17 then find A : B : C : D. Solution: A : B = 3 : 4 B : C = 8 : 10 C : D = 15 : 17 A : B : C : D = 3×8×15 : 4×8×15 : 4×10×15 : 4×10×17 = 9 : 12 : 15 : 17 www.testbag.com

8. TestBag TestBag Ratios are compounded by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. 1.1 COMPOUND RATIO Example : The ratio compounded of the four ratios i.e. 2 : 3, 5 : 11, 18 : 7 and 21 : 4 is Solution : = 45 : 11 2 × 5 × 18 × 21 45 = ——————— = —— 3 × 11 × 7 × 4 11 www.testbag.com

9. TestBag TestBag A ratio, in the simplest form, is when both terms are integers, and when these integers are prime to one another. We may multiply or divide both terms of a ratio by the same number without affecting the value of the ratio. However, addition or subtraction of same numbers will affect the value of the ratio. To express the ratio of two quantities, they must be expressed in the same units. Example : To find the ratio of 8 inches and 6 feet . First, we change 6 feet into inches, and then will find the ratio. 1 9 1 foot = 12 inches, so 6 feet = 12 x 6 = 72 inches. The ratio Or, 1 : 9 8 —— = — 72 www.testbag.com

10. TestBag TestBag 1.2 INVERSE RATIO If a : b be the given ratio, then 1/a : 1/b or b : a is called its inverse or reciprocal ratio. Example : A, B, C and D are four quantities of the same kind such That A : B = 3 : 4, B : C = 8 : 9, C : D = 15 : 16. Find the Ratio A : D. Solution : ∴ ∴ A : D = 5 : 8 A A B C 3 8 15 5 A 3 B 8 C 15 — = — × — × — = — × — × — = — — = —, — = —, — = — D B C D 4 9 16 8 B 4 C 9 D 16 www.testbag.com

11. TestBag TestBag 1.3 COMPARISON OF RATIOS a c We say that (a : b) > (c : d)  — > — b d Some other Ratios are :- • Duplicate Ratio of (a : b) is (a2 : b2) • Sub-duplicate Ratio of (a : b) is (√a : √b). • Triplicate Ratio of (a : b) is (a3 : b3). • Sub-triplicate Ratio of (a : b) is (a1/3 : b1/3). a c c + d a + b • If — = —, them ——— = ——— (Componendo and Dividendo) b d a - b c - d www.testbag.com

12. TestBag TestBag A proportion is a statement where two ratios are equal. If, m/n = p/q, it means m/n is in proportion with p/q and can be written as m : n :: p : q, where m and q extreme left and right parts are known as extremes and the middle parts n and p are known as means or interim. What is a proportion? If four quantities be in proportion, the product of the extremes is equal to the product of the means or interims. m p or, — = — or m : n : : p : q n q ∴ m × q = n × p Or m × q = Product of extremes And n × p = Product of interims (means) Product of extremes = Product of interims (means) www.testbag.com

13. TestBag TestBag PROPORTION Means the equality of 2 ratios. In symbols, it is a/b = c/d or Means a : b = c : d 1st term 4th term 2nd term 3rd term Extremes Note : b and d must not be zero www.testbag.com

14. TestBag TestBag Example: Find x, a : b = c : d x : 5 = 15 : 25 The product of the means equals the product of the extremes. 25x = 5(15) 25x = 75 x = 3 OR ad = bc www.testbag.com

15. TestBag TestBag Identify the MEANS and the EXTREMES and look for the missing term (x). 1. 3 : x = 9 : 21 9x = 63 x = 7 2(90-x) = 8x 180–2x = 8x -2x- 8x = -180 -10x = -180 x = 18 2. (x + 2) : 8 = (3x – 7) : 16 8(3x-7) = 16(x+2) 24x-56 = 16x+32 24x-16x = 32+56 8x = 88 x = 11 x 2 3. = 8 90 - x www.testbag.com

16. TestBag TestBag 2.1 FOURTH PROPORTIONAL If a : b = c : d, then d is called the fourth proportional to a, b, c. Example : Find the fourth proportional to the numbers 6, 8 and 15. Solution : If x be the fourth proportional, then 6 : 8 = 15 : x 8 × 15 ∴ x = ——— = 20 6 www.testbag.com

17. TestBag TestBag 2.2 THIRD PROPORTIONAL If a : b = c : d, then c is called the third proportional To a and b. Example : Find the third proportional to 15 and 20. Solution : Here, we have to find a fourth proportional to 15, 20 and 20 20 × 20 ∴ x = ——— = — = 26— If x be the fourth proportional, we have 15 : 20 = 20 : x 15 2 80 3 3 www.testbag.com

18. TestBag TestBag 2.3 MEAN PROPORTIONAL — Mean proportional between a and b is √ab . Example : Find the mean proportional between 3 and 75. Solution : If x be the required mean proportional, we have 3 : x : : x : 75 ——— ∴ x = √3 x 75 = 15 www.testbag.com

19. TestBag TestBag When any two quantities are related to each and if one increase or decreases, the other also increases or decreases to same extent, then they are said to be directly proportional to each other. Two variables a and b are directly proportional if they satisfy a relationshipof the form a = kb, where k is anumber.. 2.4 TYPES OF PROPORTIONS A.) Direct Proportions Example : If 5 balls cost Rs 8, what do 15 balls cost ? Solution : 5 balls : 15 balls : : Rs 8 : required cost 15 x 8 ∴ The required cost = Rs ——— 5 = Rs 24 www.testbag.com

20. TestBag TestBag When the two quantities are related to each other in a such way that if one increase, the other decreases to the some extent and vice-versa, then theyare said to be inversely proportional to each other. Twovariables a and b are indirectly proportional if they satisfya relationship of the form k = ab, where k is a number. B.) Inverse Proportion Example : If 15 men can reap a field in 28 days, in how many Days will 10 men reap it? Solution : 1 1 ∴— : — : : 28 ─ The required number of days 15 x 28 15 10 ∴ The required number of day = ———— Or, 10 : 15 : : 28 ─ The required number of days 10 = 42 www.testbag.com

21. TestBag TestBag Thanking you and Good bye… www.testbag.com