Ratio and Proportion With solved examples
Ratio
A ratio is a comparison of two quantities by division. It shows how one quantity compares with another in magnitude. It is written as a:b or a/b, which tells us what part a is of b. Here a is called the antecedent and b is called the consequent.
Types of ratio
When two or more ratios are multiplied together, the result is called a compound ratio. The compound ratio of a:b and c:d is ac : bd
The square of a ratio. Duplicate ratio of a:b = a2 : b2
The cube of a ratio. Triplicate ratio of a:b = a3 : b3
The square root of a ratio. Sub-duplicate ratio of a:b = a1/2 : b1/2
The cube root of a ratio. Sub-triplicate ratio of a:b = a1/3 : b1/3
If a:b is the given ratio, then b : a is its inverse ratio.
Properties of ratio
- In a ratio, the quantities compared must be of the same kind.
- The ratio of two quantities tells how many times one quantity is contained in the other.
Important facts
To find A:C, use A/B × B/C = A/C, so A:C = xp : yq.
To find A:B:C: A:B:C = xp : yp : yq
To find A:D: A:D = xpm : yqn
To find A:B:C:D: A:B:C:D = xpm : ypm : yqm : yqn
Solved examples on ratio
Ex.1Find the ratio compounded of the four ratios 4:3, 9:13, 26:5 and 2:15.
Required ratio = (4 × 9 × 26 × 2)/(3 × 13 × 5 × 15) = 1872/2925 = 16/25
Ans: 16 : 25
Ex.2If A:B = 3:4 and B:C = 8:10, find A:B:C.
A:B = 3:4 and B:C = 8:10
Multiply A:B by 8 (the first term of B:C) and B:C by 4 (the second term of A:B) so that B is the same in both.
A:B:C = 3 × 8 : 4 × 8 : 4 × 10 = 24 : 32 : 40 = 3 : 4 : 5
Ans: 3 : 4 : 5
Ex.3If A:B = 3:4, B:C = 8:10 and C:D = 15:17, find A:B:C:D.
A:B:C:D = 3 × 8 × 15 : 4 × 8 × 15 : 4 × 10 × 15 : 4 × 10 × 17
= 360 : 480 : 600 : 680 = 9 : 12 : 15 : 17
Ans: 9 : 12 : 15 : 17
Proportion
If four quantities are in proportion, the product of the extremes is equal to the product of the means: if a:b :: c:d, then a × d = b × c.
If a:b :: b:c, then c is called the third proportional.
a/b = b/c, so c = b2/a
If a:b :: c:d, then d is called the fourth proportional.
a/b = c/d, so d = bc/a
If a:b :: b:c, then b is called the mean proportional between a and c.
a/b = b/c, so b2 = ac and b = √(ac)
Three quantities a, b, c of the same kind are in continued proportion when a:b :: b:c. The middle number b is the mean proportional between the two extremes.
a/b = b/c, so b2 = ac
Componendo and Dividendo
If a:b is equal to c:d, that is a/b = c/d, then:
- Componendo rule: (a + b)/b = (c + d)/d
- Dividendo rule: (a − b)/b = (c − d)/d
- Componendo and Dividendo: (a + b)/(a − b) = (c + d)/(c − d)
Direct proportion
Two quantities are so related that if one of them is multiplied by any number, the other is also multiplied by the same number.
Ex.4If 5 balls cost Rs. 10, what do 15 balls cost?
If the number of balls is increased 2 or 3 times, the price is also increased 2 or 3 times.
5 balls : 15 balls :: Rs. 10 : required cost
Required cost = (15 × 10)/5 = Rs. 30
Ans: Rs. 30
Inverse proportion
Two quantities are so related that if one of them is multiplied by any number, the other is divided by the same number, and vice versa.
Ex.5If 15 men can reap a field in 28 days, in how many days will 10 men reap it?
If the number of men increases n times, the number of days decreases n times.
1/15 : 1/10 :: 28 : required number of days
or 10 : 15 :: 28 : required number of days
Number of days = (15 × 28)/10 = 42
Ans: 42 days
The Rule of Three
The method of finding the 4th term of a proportion when the other three terms are given is called simple proportion or the Rule of Three.
- Denote the quantity to be found by the letter x and set it down as the 4th term.
- Of the three given quantities, set down for the 3rd term the one which is of the same kind as the quantity to be found.
- Check whether the quantity to be found will be greater or less than the 3rd term. If greater, make the greater of the two remaining quantities the 2nd term and the other the 1st term. If less, make the lesser quantity the 2nd term and the greater the 1st term.
- Required value = (product of the means)/(1st term).
Ex.6If 15 men can reap a field in 28 days, in how many days will 5 men reap it?
Method 1 (Rule of Three)
Step 1. _ : _ :: _ : required number of days
Step 2. _ : _ :: 28 : x
Step 3. The required number of days will be more, since 5 men will take more time than 15 men. Therefore 5 : 15 :: 28 : x
Step 4. x = (15 × 28)/5 = 84 days
Method 2 (Man-Day formula)
M1D1 = M2D2
15 × 28 = 5 × x, so x = (15 × 28)/5 = 84
Ans: 84 days
Compound proportion (Double Rule of Three)
Ex.7If 30 men working 7 hrs a day can do a piece of work in 18 days, in how many days will 21 men working 8 hrs a day do the same piece of work?
Method 1
21 men : 30 men :: 18 days : required number of days (fewer men take more days)
8 hrs : 7 hrs (more hours per day take fewer days)
Required days = (product of means)/(product of first terms) = (30 × 7 × 18)/(21 × 8) = 3780/168 = 22.5 days
Method 2
M1D1H1 = M2D2H2
30 × 18 × 7 = 21 × x × 8
x = (30 × 18 × 7)/(21 × 8) = 22.5 days
Ans: 22.5 days
Proportional division
Proportion can be applied to divide a given quantity into parts.
Ex.8Divide Rs. 1350 into three shares proportional to the numbers 2, 3 and 4.
Sum of the ratio terms = 2 + 3 + 4 = 9
1st share = 1350 × 2/9 = Rs. 300
2nd share = 1350 × 3/9 = Rs. 450
3rd share = 1350 × 4/9 = Rs. 600
For more examples of proportional division, visit the post on partnership.
Ans: Rs. 300, Rs. 450 and Rs. 600
Comments
Post a Comment