Today’s competitive exams are much more dynamic in terms of questions and their patterns. Quantitative aptitude is a very important section from any competitive exam point of view and “Ratio and Proportion” is considered to be the most important topic in quantitative aptitude. Every year a good number of ratio and proportion questions comes from this topic. Concept and formulas from ratio and proportion can be applied to solve other topics like “Time and Distance”, “Allegation and Mixture”. Aspirants can definitely expect 3-5 Ratio and Proportion questions in exams like IBPS PO, SBI PO, IBPS RRB, IBPS Clerk, SBI Clerk, IBPS RRB, SSC CGL, SSC CGL, SSC MTS, NICL AO, LIC AAO, SBI Associate Clerk, SBI Associate PO, and others.

In this article step-by-step we will discuss important concepts tricks and tips to solve “Ratio and Proportion” questions in Quantitative aptitude. Using these simple techniques you can solve any types of Ratio and Proportion questions in minimum time. In our previous article, we have discussed “How to solve Blood Relation reasoning questions quickly”.

For easy understanding of this topic, we explained a few “Ratio and Proportion” questions from recent competitive exam.

### What is Ratio?

Ratio is a unitless quantity which represents the number of times the value of one quantity contains another quantity of the same kind. In other words, the ratio of two quantities is equivalent to the fraction that one quantity is of the other or it is a magnitude by which one quantity is multiple of others.

Ratio is represented as “a to b”, “a:b ” or “a/b”. The first term( a ) is called antecedent and the second term( b ) is called consequent.

### Properties of Ratio

- Ratio does not have any dimension. It is a mere number. It can be defined as a quotient of two numbers. Since the quotient obtained on dividing one value by another value of the same kind is an abstract number, the ratio of the two value of the same kind is an abstract number. thus the ratio between 3kg and 5 kg is 3:5.
- A ratio which is a fraction cannot be altered by multiplying or dividing both its term by the same number. Thus 2:3 is the same as 6:9.

### Compounded Ratio

Ratios are compounded by multiplying together antecedents and consequents to form a new antecedent and a new consequent.

For example,

The compounded ratio of a:b and x:y is ax:by.

#### Duplicate and Sub-duplicate Ratio

When any ratio compounded with itself, the resulting ratio is called a duplicate ratio.

If the ratio is a:b then:

The duplicate ratio of a:b is

Similarly, the square root of any ratio is known as sub-duplicate ratio.

The sub-duplicate ratio of a:b is

#### Triplicate and Sub-triplicate ratio

When any ratio compounded with itself thrice, the resulting ratio is called triplicate ratio.

If the ratio is a:b then:

Triplicate ratio of a:b is

Similarly, cube root of any ratio is known as sub-triplicate ratio.

Sub-duplicate ratio of a:b is

#### Inverse Ratio

If the ratio is a:b, then 1/a : 1/b or b:a is called inverse or reciprocal ratio of a:b.

- When antecedent equals to consequent, the ratio is called the ratio of equality, such as 1:1.
- When antecedent is greater than consequent, the ratio is called the ratio of greater inequality. Such as 5:4.
- When antecedent is lesser than consequent, the ratio is called the ratio of lesser inequality. Such as 4:5.

#### How to find Common Ratio

If the ratio between the first and the second quantity is a:b and the ratio between the second and third quantity is c:d, then the ratio among first, second and third quantity is ac:bc:bd.

The above ratios can be written as:

Writing b at the space above d and write c at the space below a.

Multiplying columns we get the common ratio i.e. ac:bc:bd

For 3 or more quantities step by step, we can find the common ratio by a simple trick.

For example,

If there are four quantities and the respective ratios between first and second, second and third and third and fourth are a:b,c:d,e:f.

##### Step#1

In the first step write down the antecedent of the second ratio below the consequent of the first ratio. Follow the same method for second and third ratios and so on.

##### Step#2

In every row, fill every space before antecedent with the value of antecedents and space after consequent with the value of consequent.

##### Step#3

Finally, multiply the column vertically to get the common ratio.

#### Problems on Ratio

**E.g 01. **In an exam, the total marks obtained by both A and B is 378. The ratio between the marks obtained individually is 5:9. What is the total marks of B?

Sol. let’s take A got 5x and B got 9x. Then the total marks obtained by them is 5x+9x=11x.

11x=378 so, x=27

Therefore A got 135 and B got 243.

**E.g 02.** Divide Rs 625 among A, B and C such that A gets 2/9 of B’s share and C gets ¾ of A’s share.

Sol. A:B=2:9 and C:A=3:4

then A:B:C is 8:36:6 or 4:18:3. therefore A got Rs100, B got Rs 450 and C got Rs 75.

**E.g 03.** A can finish a work in 7 days. B is 40% more efficient than A. Find the number of days it takes B to do the same piece of work.

Sol. let’s efficiency of A is 100 and efficiency of B is 140. As B is more efficient, it is clear that B will complete the work in fewer days. So, the number of days required by B is

**E.g 04.** Rs 425 is divided among 5 men, 6 women and 4 boys such that the share of a man, a woman and a boy may be in the ratio of 8:4:9. What is the share of a woman?

Sol. The ratio of shares of group of men, woman and boys = 5×8 : 6×4 : 4×9 = 40 : 24 : 36

Share of one woman = 102/6 = Rs 17.

### What is Proportion?

If two ratios are equal in value, they are in proportion. If the ratio a : b is equal to the ratio c : d, then a, b, c, d are in proportion.

Using symbols we write as a : b = c : d or a : b :: c : d.

For example,

Consider two ratios 5:25 and 8:40.

Since 5 is one-fifth of 25 and 8 is one-fifth of 40, then two ratios are equal. Hence 5,25,8 and 40 are said to be in proportion.

We can write proportion as [5:25::8:40] or [5/25=8/40]

#### Properties of Proportion

- If a,b,c and are in proportion, a and d are called extremes and b and c are called means.

For example,

Let’s the number 6,18,8 and 24 are in proportion. The first and fourth term i.e., 6 and 24 are called extremes, and the second and the third term i.e.,18 and 8 are called means.

- If four quantities are in proportion, the product of the extremes equals the product of the means.

In the above example, we have 6 and 24 as extremes and 18 and 8 as means.

Since the multiplication of 6 and 24 i.e. 144 equals to the multiplication of 18 and 8. Therefore multiplication of extremes equals to the multiplication of means.

- If a,b,c and are in proportion,then

- a/b=c/d=a+c/b+d
- Invertendo- If a/b=c/d, then b/a=d/c
- Alterendo- If a/b=c/d, then a/c=b/d
- Componendo- If a/b=c/d, then a+b/b=c+d/d
- Dividendo- If a/b=c/d, then a-b/b=c-d/d
- Componendo & Dividendo- If a/b=c/d, then a+b/a-b=c+d/c-d

#### Continued Proportion

Three quantities are said to be in proportion when the ratio of first to the second quantity is equal to the ratio of second to the third.

If a,b and c are in continued proportion, then a/b=b/c.

#### Mean Proportion

If a,b and c are in continued proportion, then the second term is called the mean proportional between first and second.

If a,b and c are in continued proportion, then a/b=b/c , b∧2=ac , b=√ac.

For example, in continued proportion “9:6:6:4”, 6 is the mean proportional between 9 and 4, and 4 is the third proportional to 9 and 6.

#### Direct Proportion

If an increase in one quantity results in an increase in the other quantity and decrease in one quantity results in a decrease in the other quantity. Then both the quantities are in direct proportion. If the first quantity is increased or decreased by a fraction then the second quantity also increased or decreased by the same fraction.

If two quantities x and y are in direct proportion, then the relation is represented as

x=ky. where k is increasing or decreasing factor or proportionality constant. If x1, x2 and y1, y2 are two variables of x and y, then x1/y1=x2/y2.

#### Examples of Direct Proportion

In mathematics and physical science, there are innumerous examples where you find direct proportion in several quantities.

- Men and work are in direct proportion, as more the men available, the work will be done more. Same as in work and days, more work requires more number of days to complete.
- Chapter like time and distance, there is also a direct proportion in distance and time, because more the distance traveled, the time taken will be more (if speed remains constant).
- Purchase of goods and the amount spent are in direct proportion, as the purchase of more goods will cost more money.

#### Problems on Direct Proportion

**E.g 05.** In a given time, 15 person makes 111 toys. How many person should be employed to make 148 toys at the same time?

Sol. As person and number of toys are in direct proportion. Then the ratio of the person would be the same with the ratio of work.

Therefore, No. of person needed/15=148/111

No. of person=20

**E.g 06.** If 168 mangoes bought for rupees 12, how many mangoes can be bought for rupees 6?

Sol. As the number of mangoes bought is directly proportional to the cost of mangoes. then the ratio of the number of mangoes equals to the ratio of the cost.

Therefore, No. of mangoes bought/168=6/12

No. of mangoes is=84.

#### Inverse Proportion

If an increase in one quantity results in a decrease in the other quantity and decrease in one quantity results in an increase in the other quantity. Then both the quantities are in inverse proportion. If the first quantity is increased or decreased by a fraction then the second quantity also decreased or increased by the same fraction.

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 they are inversely proportional. The relation between “x” and “y” is represented as xy=k. where k is increasing or decreasing factor or proportionality constant.

If x1, x2 and y1, y2 are two variables of x and y. So x1y1=x2y2, when x and y are in inverse proportion.

#### Examples of Inverse Proportion

Like direct proportion inverse proportion is found in lots of physical and mathematical quantity around us.

- Population and Quantity of food consumption are in Inverse Proportion, because if the population increases, the food availability decreases.
- Number of men and time are in inverse proportion, as more the men available, the work will be done in less time.
- Speed and Time are in Inverse proportion, because the higher the speed, the lower is the time taken to cover a distance.

#### Problems on Inverse Proportion

**E.g 07.** If 30 men can do a piece of work in 25 days, in what time 15 man do another piece of work.

Sol. In this question ratio of the number of men is inversely proportional to the ratio of the number of days needed to complete the work. As more men can complete a work in fewer days.

Hence, No. of days required/25=30/15. No. of days required=30*25/15.

**E.g 08.** To cover a distance A takes 30 min and B takes 45 min. If the speed of A is 24 kmph. Find speed of B?

Sol.

**Method 01**– let’s take the distance is X. A covers X distance in 30 min and B covers this distance in 45 min. So, speed of A= (X*60)/30………..(1) speed of B= (X*60)/45…………(2)

Speed of A=24 kmph= (X*60)/30, X=12. Putting the value of X in 2nd equation we get the speed of B is 16 kmph.

**Method 02**– Speed of A/ Speed of B= Time taken by B/ Time taken by A

24/Speed of B=45/30 hence, Speed of B=30*24/45=16 kmph.

#### Miscellaneous Example

Here are some more ratio and proportion questions from different competitive exam.

**E.g 09.** What must be added to the two numbers that are in a ratio of 3:4, So that they become in the ratio 4:5?

Sol. In this problem, both numerator and denominator increases by the same value. Let’s say the value is X. then 3+X/4+X=⅘, 15+5X=16+4X. Hence X=1.

**E.g 10.** An employer reduces the number of his employees in the ratio 5:4 and increases their wages in the ratio of 12:13. State whether his bill of total wages increases or decreases, and in what ratio?

Sol. Ratio of employee 5:4 ratio of wages 12:13

We know that the total bill= wage per person x no. of total employees

Therefore, the ratio of change in bill=5×12:4×13=60:52=15:13

The ratio shows that there is a decrease in the bill.

**E.g 11.** A hound pursues a hare and takes 6 leaps for every 8 leaps of the hare, but 3 leaps of the hound are equal to 5 leaps of the hare. Compare the rates of the hound and the hare.

Sol. 3 leaps of the hound = 5 leaps of the hare

6 leaps of the hound = 30/3 leaps of the hare

Then the rate of hound : rate of the hare=10:8=5:4

**E.g 12.** In 30 liters of milk, the ratio of milk to water is 4:1. Another 3 liters of water is added to the mixture. What will be the ratio of milk to water in the new mixture?

Sol. Milk in 30 liters of milk=30x(⅘)=24 liters and water=6 liters

After adding 3 liters of waters, total quantity of water=6+3=9

New ratio in the milk is=24:9=8:3

**E.g 13.** Two container A and B contain a mixture of Acid and Water in the ratio of 5:6 and 4:1. If both the containers are mixed in the proportion of 5:3. Find the ratio of Acid and water in the new mixture?

Sol. Let’s consider there are 5 liters of mixture A mixed with 3 liters of mixture B.

In 5 liters of mixture A, acid= 5x(5/11) and water=5x(6/11)

In 3 liters of mixture B, acid=3x(⅘) and water=3x(⅕)

Required ratio=(25/11+12/5):(30/11+3/5)=125+132:150+33=257:183

Finally, we have learned all the important concept tricks and tips to solve ratio and proportion questions for the upcoming exams.

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