Concept of Ratio and Proportion

Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. The ratio and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science.

• Equivalent Ratios Let us divide a Pizza into 8 equal parts and share it between Ram and Sam in the ratio 2:6. The ratio 2:6 can be written as 2/6;2/6 = 1/3 We know that 2/6 and 1/3 are called equivalent fractions. Similarly we call the ratios 2:6 and 1:3 as equivalent ratios.

• From a given ratio x : y, we can get equivalent ratios by multiplying the terms ‘x’ and ‘ y ‘by the same non-zero number.
  For example
  
  1 : 3 = 2 : 6 = 3 : 9
  4 : 5 = 12 : 15 = 16 : 20
• Ratio and Proportion Problems and Solutions
  
  Example 1: Write any 4 equivalent ratios for 4 : 3.
  Sol: Given Ratio = 4 : 3. The ratio in fractional form = 4/3, we can get equivalent ratios by “4”and “3” by 2, 3, 4, 5 and get the equivalent fractions of 4/3 are 8/6, 12/9, 16/12, 20/15,
• ∴ The equivalent ratios of 4 : 3 are 8 : 6, 12 : 9, 16 : 12, 20 : 15

Types of Ratio:

• If a : b is a ratio then:
   Duplicate ratio of (a : b) is (a² : b²).
   Sub-duplicate ratio of (a : b) is (a^1/2 : b^1/2).
   Triplicate ratio of (a : b) is (a³ : b³).
   Sub-triplicate ratio of (a : b) is (a^1/3 : b^1/3).

• Example 3: What is the duplicate ratio of 2 : 3?
  Sol: Duplicate ratio of 2 : 3 = 2² : 3² = 4 : 9.
   
• Example 4: Triplicate ratio of two numbers is 27 : 64. Find their duplicate ratio.
  Sol: Triplicate ratio of two numbers is 27 : 64, so numbers should be 271/3 : 641/3 So numbers are in the ratio 3 : 4. So duplicate ratio of 3 : 4 = 3² : 4² = 9 : 16.
   
• Example 5: The ratio of two numbers is 25 : 36. Find their sub duplicate ratio.
   
  Sol: Sub duplicate ratio of 25 : 36 = 25^1/2 : 36^1/2 = 5 : 6.

PROPORTION

Proportion is represented by the symbol ‘= ‘or ‘:: ‘
If the ratio a : b is equal to the ratio c : d, then a, b, c, d are said to be in proportion.
Using symbols we write as a : b = c : d or a : b :: c : d

• When 4 terms in proportion, then the product of the two extremes (i.e. the first and the fourth value) should be equal to the product of two middle values (i.e. the second and the third value)

• Example 6: Prove that 16 : 12 and 4 : 3 are in proportion.
  Sol: The product of the means = 12 × 4 = 48. The product of the extremes = 16 × 3 = 48
  As Product of Means = Product of Extremes ∴ 16 : 12, 4 : 3 are in proportion.
   
• Example 7: Find the missing number in 3 : 4 = 12 : ____
  Sol: Let the missing number is “a”. We know that, Product of means = Product of extremes.
  Therefore 3 × a = 4 × 12; By dividing both sides by 3, we get the missing term = (4 × 12)/3 = 16
   
• Example 8: Taking 4 and 16 are means, write any two proportions.
  Sol: Given 4 and 16 are means. So, __: 4 = 16: __
  The product of Means is 4 × 16 = 64. Hence the product of Extremes must also be 64
  64 can be written as 4 × 16 or 2× 32 etc. Two proportions are 2: 4:: 16 : 32 and16 : 4 :: 16 : 4.

FOURTH PROPORTIONAL:
If a : b = c : d, then d is called the fourth proportional to a, b, c.

• Example 9: Find the fourth proportional of the numbers 12, 48, 16.
  Sol: Let fourth proportional is x. Now as per the concept above the product of extremes should be equal to the product of the means → 12/48 = 16/x → x = 64.

THIRD PROPORTIONAL: a : b = c : d, then c is called the third proportion to a and b.

• Example 10: If 2, 5, x, 30 are in proportion, find the third proportional “x”.
  Sol: Here x is third proportional. According to the concept 2/5 = x/30 → x = 12.
• MEAN PROPORTIONAL: Mean proportional between a and b is √ab .

• Example 11: Find the mean proportional of the numbers 10 and 1000.
  Sol: Mean proportional between a and b is √ab. Let the mean proportional of 10 and 1000 be x.
  So x = √10x1000 = √10000 = 100.

• CONTINUED PROPORTION a, b, c are in Continued Proportion if a : b = b : c. Here b is called the Mean Proportional and is equal to the square root of the product of a and c.{b² = a × c → b = √ac}

• a/b = b/c = c/d etc., then a, b, c, d are in Geometric Progression.
  Let a/b = b/c = c/d = k, then, c = dk; b = ck and a = bk
  Since c = dk, b = dk × k = dk2 and a = bk = dk2 × k = dk3, implying they are in Geometric Progression.
  If the three ratios, a : b, b : c, c : d are known, we can find a : d by the multiplying these three ratios
  a/d = a/b × b/c × c/d

• If a/b = c/d= e/f , then each of these ratios is equal to (a+c+e)/(b+d+f)
• If a/b = c/d, then b/a = d/c(Invertendo)
• If a/b = c/d, then a/c = b/d (Alternendo)
• If a/b = c/d , then (a+b)/b = (c+d)/d(Componendo)
• If a/b = c/d, then (a-b)/b = (c-d)/d (Dividendo)
• If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d), (Componendo & dividendo)

• Example 12: If a : b = 2 : 5, then find the value of (3a + 4b) : (5a + 6b).
  Sol: Let a = 2x & b = 5x. Then (3a + 4b): (5a + 6b) = (3 × 2x + 4 × 5x) : (5 × 2x + 6 × 5x) → 26x:40x = 13 : 20.

• DIRECT VARIATION Two quantities “x” and “y” are said to be in direct variation if an increase in one quantity results in increase in the other quantity and decrease in one results in decrease in the other quantity. If two quantities vary always in the same ratio, then they are in direct variation.

• Examples for Direct Variation:

  1. Distance and Time are in Direct Variation, because more the distance travelled, the time taken will be more (if speed remains the same).
  2. Principal and Interest are in Direct Variation, because if the Principal is more, the Interest earned will also be more.
  3. Purchase of Articles and the amount spent are in Direct Variation, because purchase of more articles will cost more money. If two quantities “x” and “y” vary directly in such a way that x/y remains constant and is positive, and this constant is called the constant of variation. If x α y that means x = py where p is proportionality constant x/y = p, then ratio of any two values of “x” is equal to the ratio of corresponding values of “y” .Then x₁/x₂=x₂/y₂

  INVERSE VARIATION:

  If two quantities “x” and “y” are such that an increase or decrease in “x” leads to a corresponding decrease or increase in “y” in the same ratio, then we can say they vary indirectly or the variation is inverse. Suppose 6 men can do a piece of work in 18 days, then 12 men can do the same job in 9 days. That means if we double the number of men, then number of days get halved. That means there is inverse relation between number of men and number of days.
  In general, when two variables x and y are such that xy = k where k is a non-zero constant, we say that y varies inversely with x. In notation, inverse variation is written as y α 1/x → y = p/x, where p is constant of proportionality → xy = p. So x₁y₁ = x₂y_2^.

  Examples for Inverse Variation:

  1. Work and Time are in Inverse Variation, because more the number of the workers, lesser will be the time required to complete a job.
  2. Speed and Time are in Inverse Variation, because higher the speed, the lower is the time taken to cover a distance.
  3. Population and Quantity of food are in Inverse Variation, because if the population increases, the food availability decreases.

   

  Example 13: Suppose that y varies inversely as x and that y = 12 when x = 6.
  a) Form an equation connecting x and y.
  b) Calculate the value of y when x = 18.

  Sol: x and y are in inverse proportion. So x₁y₁= x₂y₂ → 6 × 12 = 18 × y → y = 4