RATIO,PROPORTION & VARIATION

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RATIO & PROPORTION

GETTING STARTED

☛ A fraction a/ b can be expressed as a:b

☛ Generally, ratio is used as a tool to compare two quantities.

 Let us say, in a jar there are 5L milk and 4 L water. We say the ratio of milk to water in the Jar is 5:4.

☛ The ratio remains unaffected when it is multiplied by any natural number.

eg. 5:4 =5 x 4: 4 x 4 = 20:16

 =5 x 16: 4 x 16 = 80:64

=5 x 17: 4 x 17= 85:68

☛ Proportion :
When two ratios a:b and c:d are equal, we say that they are in proportion.
They are denoted as a:b::c:d
a/b = c/d

Solving,
ad = bc

The third term c in the proportion a:b::c:d is known as ‘Third Proportional‘.

The fourth term d in the proportion a:b::c:d is known as ‘Fourth Proportional‘.

 AGE PROBLEMS

GETTING STARTED

 ☛ If a person’s age is x years, then after n years, his age will be x+n years

☛ Age difference two persons always remains constant.

Eg. Ram is 16 years today and his mother is 38 years old. What will the age difference between then after 10 years?

Soln:
Ram is 16 years and his mother is 38 years old today. Difference in their ages is 22 years.

It means Ram’s mother was 22 years when he was born.

The difference will be always present.

10 years hence, Ram will be 26 years and his mother will be 48 years. Difference in ages = 48-26= 22 years

Ratio of ages

☛ The ratio of ages between two persons keeps changing every year.

☛ All the age problems can be solved by converting the problem into word problem in one variable

Eg. The ratio of Ram and Rahim’s age is 2:3 today. After 9 years, their ages will be in the ratio 7:9.
Find their present ages.
Soln:
Let Ram’s age be 2x. Thus Rahim’s present age is 3x

After 9 years, their ages will be 2x+9 and 3x+9 respectively.

(2X+9)/(3x +9) = 7/9

18x+81 = 21x+63

3x = 18

X =6

Hence, Ram’s age is 12 years and Rahim’s age is 18 years.

TIP: The best way to solve the age problems is to apply options given as per the conditions mentioned in question.

 VARIATION 

GETTING STARTED

Variation is one of the most fundamental concept which will be a building block for other concepts.

☛ Suppose you go to a nearby shop and see that 10 books cost Rs. 400. You want to buy 21 such books. What would be the cost of 21 such books?

Since 10 books cost Rs. 400,

Cost of 1 book should be 400/10 = Rs.40

Cost of 21 books = Cost of 1 book x 21

                               = 40 x 21

                               = Rs.840

 

☛ The method of finding the value for n units given that you know the value for 1 unit is known as ‘Unitary Method’.

Here, we noticed that with increase in number of books, total money spent also increased. This is an example of ‘Direct Variation’. With an increase of X, there is also an increase of Y.

☛ Another example can be angle traced by Clocks.

 Minute hand of a clock traces 6° in one minute. How much angle does it cover in 40 minutes?

More minutes, more angle.

Hence in 40 minutes, minute hand would cover 40 x 6° = 240°

                  

      

To find the value of ?, we cross multiply,

? = 40 x 6/1 = 240°

 

 Inverse Variation

There are cases where the value of Y decreases with the increase in value of X.

☛ Let us take an example of 10 men working on a project and completing it in 5 days. If the same work is to be done by 20 men, how much time will they take?

☛ By logic, we know, more men, less the time taken. So, this is a case of inverse variation.

Considering 1 person completes 1 unit of work in a day, 10 men working 5 days complete 50 units of work. Let us say the entire job consists of 50 smaller jobs.

So, 1 man will complete the entire job (with 50 units of work) in 50 days.

 

☛ 20 men will complete 20 units of work in 1 day. Hence, they will take 50/20 = 2.5 days to complete the entire work.

 

 ? = 10 x 5/20 = 2.5

☛ If we notice the arrow marks in direct variation and inverse variation, direct variation had cross multiplication of values while inverse variation, we multiply values in the same row.

The questions will not be directly asked on this topic but it is the foundation for all the topics 

PRACTICE PROBLEMS

3

Ratio, Proportion & Variation

1 / 40

Rakesh is elder than Roshan by 12 years. 8 years ago, the ratio of their ages was 5:3.What is the present age of Roshan?

2 / 40

A gives 15 chocolates to B and notices that the ratio of chocolates with A and B would get reversed. Find the difference in number of chocolates with them.

3 / 40

If a:b = b:c = 4:7, find the value of a:c

4 / 40

990 is divided in such a way that half of the first part equals 1/3rd of the second part which in turn equals 1/4th of the third part. Find the value of 1st part.

5 / 40

840 is divided into 3 parts in such a way that 21 times first part equals 24 times the second part which in turn equals 28 times the third part. What is the first part?

6 / 40

Two numbers are in the ratio 7:9. What number must be added to each number so that their ratio becomes 5 : 6?

7 / 40

Suresh and Ganesh have toffees in the ratio 25:17. If Suresh gives 44 of them to Ganesh, both of them have equal number of toffees. Find the number of toffees initially with Ganesh.

8 / 40

If a:b is 6:7, find the value of (4a+3b):(3a+6b)

9 / 40

The test scores of Ganesh and Vinayak are in the ratio 5:6. The sum of their scores is 143. What is Vinayak's score?

10 / 40

If 42 men can complete a work in 14 days, how many days will 28 men take to complete the same work?

11 / 40

10 carpenters make 10 furniture in 10 days. How many furniture will 20 carpenters make in 20 days?

12 / 40

There are 144 employees working in an office. Which of the following cannot be a ratio of male staff to the female staff in the office?

13 / 40

If x:y = 5:7 and x:y = (a-b):(a+b), find the value of a:b

14 / 40

X varies directly as Y when Z is constant and varies inversely with Z when Y is constant. The value of X is 4 when Y is 18 and Z is 5. Find the value of X when the value of Y is 63 and that of Z is 7.

15 / 40

In a hypothetical exam, the marks scored by a student is directly proportional to the number of hours put in for preparation. If he works for 10 hours he scores 45 marks. How many hours did he have to study in order to score 60 marks?

16 / 40

Six years ago, Father's age was three times his son's age. Currently he is two and a half times of son's age. After further 6 years, what will be the ratio of their ages?

17 / 40

A father said to his son, "I was as old as you are at the present at the time of your birth". If the father's age is 44 years now, the son's age 3 years back was:

18 / 40

Ratio of earnings of Suresh and Gopi are in the ratio 3:4. Their expenditure is in the ratio 9:10. Which of the following can be a ratio of their savings?

19 / 40

A 40 m building casts a shadow of length 15 m. What will be the height of a tower which casts a shadow of 24 m?

20 / 40

The present ages of three persons is in proportion of 6: 4 :9. Eight years ago, the sum of their ages was 109. Find their present ages (in years).

21 / 40

Saurav is younger than Virat by 9 years. If their ages are in the respective ratio of 3 : 5, how old is Saurav?

22 / 40

At present, the ratio between the ages of Arun and Shiv is 5 : 6. After 8 years, Arun's age will be 28 years. What is the age of Shiv at present ?

23 / 40

The sum of the present ages of a father and his son is 70 years. Five years ago, father's age was five times the age of the son. After 6 years, son's age will be:

24 / 40

Six years ago, the ratio of the ages of Raju and Arjun was 3 : 2. Four years hence, the ratio of their ages will be 5 : 4. What is Raju's age at present?

25 / 40

A man is 34 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:

26 / 40

A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 37, then how old is A?

27 / 40

Present ages of Atul and Deb are in the ratio of 6 : 5 respectively. 15 years hence, the ratio of their ages will become 9 : 8 respectively. What is Atul's present age in years?

28 / 40

A person's present age is one-fifth of the age of his mother. After 15 years, he will be one-half of the age of his mother. What is the age difference between the person and his mother?

29 / 40

A and B together have Rs. 2430. If 4/15 of A's amount is equal to 2/5 of B's amount, how much amount does B have?

30 / 40

Two numbers are respectively 50% and 80% more than a third number. The ratio of the two numbers is:

31 / 40

A sum of money is to be distributed among A, B, C, D in the proportion of 6 : 3 : 4 : 2. If B gets Rs.800 more than D, what is C's share?

32 / 40

Seats for Mathematics, Physics and Biology in a school are in the ratio 6:7:8. There is a proposal to increase these seats by 40%, 50% and 57.5% respectively. What will be the ratio of increased seats?

33 / 40

The ratio of the number of boys and girls in a college is 8 : 9. If the percentage increase in the number of boys and girls be 12.5% and 10% respectively, what will be the new ratio?

34 / 40

Salaries of Ashish and Saroj are in the ratio 3 : 4. If the salary of each is increased by Rs.4000, the new ratio becomes 10:13. What is Saroj's salary?

35 / 40

The fourth proportional to 25, 15, 55 is:

36 / 40

If 80% of a number is equal to two-third of another number, what is the ratio of first number to the second number?

37 / 40

The salaries A, B, C are in the ratio 3 : 4 : 2. If the increments of 20%, 10% and 60% are allowed respectively in their salaries, then what will be new ratio of their salaries?

38 / 40

If Rs.2261 be divided into three parts, proportional to 4/5:3/4 :2/3 , then the first part is

39 / 40

The sum of three numbers is 188 . If the ratio of the first to second is 2:3 and that of the second to the third is 4:9, then the second number is:

40 / 40

A varies inversely as the square of B. If the value of A=2, when the value of B= 4, find the value of A when B = 9

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