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

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

What is ratio? 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

TestBag TestBag Example: The ratio 5 : 9 represents 5/9, here 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. — ≠ — b a www.testbag.com

TestBag TestBag  Example: If the ratio between the first and the second quantities is 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 = —————— ×15 10 + 15 + 24 98 ∴ The second number = 30 www.testbag.com

TestBag TestBag b a 1st : 2nd = 2nd : 3rd = d 3rd : 4th = c e f  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 1st : 2nd = 2nd : 3rd = 3rd : 4th = e c a d b f 1st : 2nd : 3rd : 4th = ace : bce : bde : bdf www.testbag.com

TestBag TestBag Example : Solution: 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

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

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 foot = 12 inches, so 6 feet = 12 x 6 = 72 inches. —— = — 72 8 9 1 The ratio Or, 1 : 9 www.testbag.com

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. — = —, — = —, — = — B 4 C 9 D 16 A 3 B 8 C 15 Solution : — = — × — × — = — × — × — = — D B C D 4 9 16 8 A A B C 3 8 15 5 ∴ ∴ A : D = 5 : 8 www.testbag.com

TestBag TestBag 1.3 COMPARISON OF RATIOS We say that (a : b) > (c : d)  — > — a c 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). If — = —, them ——— = ——— (Componendo and Dividendo) a c b d a + b a - b c + d c - d www.testbag.com

What is a proportion? 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. or, — = — or m : n : : p : q m p 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

TestBag TestBag Means a : b = c : d Extremes 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

TestBag TestBag a : b = c : d ad = bc 25x = 5(15) 25x = 75 x = 3 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

Identify the MEANS and the EXTREMES and look for the missing term (x). TestBag TestBag Identify the MEANS and the EXTREMES and look for the missing term (x). 3. x 90 - x 2 8 = 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 www.testbag.com

TestBag TestBag Example : Solution : 8 × 15 ∴ x = ——— = 20 6 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 ∴ x = ——— = 20 8 × 15 6 www.testbag.com

TestBag TestBag Example : Solution : ∴ x = ——— = — = 26— 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 If x be the fourth proportional, we have 15 : 20 = 20 : x ∴ x = ——— = — = 26— 20 × 20 15 3 2 80 www.testbag.com

TestBag TestBag Example : Solution : ∴ x = √3 x 75 = 15 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

TestBag TestBag 2.4 TYPES OF PROPORTIONS A.) Direct Proportions 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 relationship of the form a = kb, where k is a number.. A.) Direct Proportions Example : If 5 balls cost Rs 8, what do 15 balls cost ? Solution : 5 balls : 15 balls : : Rs 8 : required cost ∴ The required cost = Rs ——— 15 x 8 5 = Rs 24 www.testbag.com

TestBag TestBag B.) Inverse Proportion Example : Solution : 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 they are said to be inversely proportional to each other. Two variables a and b are indirectly proportional if they satisfy a 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 : ∴ — : — : : 28 ─ The required number of days 1 1 15 10 Or, 10 : 15 : : 28 ─ The required number of days ∴ The required number of day = ———— 15 x 28 10 = 42 www.testbag.com

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