Ratio and Proportion : Quantative

Ratio and proportion is the heart of arithmetic.  If you understand this chapter properly you can solve virtually any problem in arithmetic.
If two numbers are in the ratio 2:3 means for every two units of the first number, second has 3 units.  This is a mere comparison between numbers, and actual numbers may be way bigger than these numbers.  If you multiply or divide a ratio the comparison does not change. i.e., 2:3 is same as 4:6.
If two numbers are in the ratio a:b then this ration has to be multiplied with a number K, to get actual numbers.  This K is called multiplication factor (MF)
If two ratios are equal then we say they are in proportion.  then $a:b::c:d\Rightarrow a\times d=b\times c$
 or $\frac{a}{b}=\frac{c}{d}\Rightarrow a\times d=b\times c$.

Chain Rule:

Chain rule is comes in handy when there are many variables need to compare with the given variable.  We can understand this rule by observing a practice problem.

Solved Example:
If 12 carpenters working 6 hours a day can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day?
Let us prepare small table to understand the problem.

Man	Hour	Days	Chair
12	6	24	460
12	8	36	?

Now with repect to the Chairs we need to understand how each variable is related.
If the number of men got increased (i.e., 12 to 18), do they manufacture more chairs or less chairs is the question we have to ask ourselves. If the answer is "more" then the higher number between 12, 18 will go to the numerator and other will go to denominator and vice versa. Here answer is "more"
So $460\times \frac{18}{12}$
Next we go to Hours. If the number of hours they work each day got increases then do they manufactures more chairs or less chairs? Answer is more
So $460\times \frac{18}{12}\times \frac{8}{6}$
Last, If the number of day they work increases then .... answer is more.
So $460\times \frac{18}{12}\times \frac{8}{6}\times \frac{36}{24}=1380$

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$2. The ratio between Sumit's and Prakash's age at present is 2:3. Sumit is 6 years younger than Prakash. The ratio of Sumit's age to Prakash's age after 6 years will be :a. 1 : 2b. 2 : 3c. 3 : 4d. 3 : 8Correct Option: CExplanation:Let their ages be 2x and 3x years.3x-2x=6 or x=6Sumit's age = 12 years, Prakash's age = 24 yearsRatio of their ages = 18 : 24 = 3 :4.\$

3. The salary of two friends Ramu and Raju are in the ratio 4:5. If the salary of each one increases by Rs.6000, then the new ratio becomes 48:55. What is Raju's present salary?

a) Rs.10500 b) Rs.10500 c) Rs.11500 d) Rs.12500

Answer : b) Rs.10500

Solution :

Ratio their salary is 4:5
Let the original salary of Ramu and Raju be 4k and 5k respectively.

After increasing Rs.6000, the ratio becomes 48:55

That is,

(4k+6000)/(5k+6000) = 48/55
55(4k+6000) = 48(5k+6000)
220k+330000 = 240k+288000
20k= 42000

We have to find the original salary of Raju; that is, 5k.

If 20k = 42000 then 5k = 10500.

Hence the required answer is Rs.10500

4. The number of candidates writing three different entrance exams is in the ratio 4:5:6. There is a proposal to increase these numbers of candidates by 40%, 60% and 85% respectively. What will be the ratio of increased numbers?

a) 14:15:16 b) 12:15:19 c)13:19:21 d) none of these

Answer : d) none of these

Solution :

Given ratio of number of candidates is 4:5:6

Let the number of candidates for 3 exams be 4k, 5k and 6k respectively.

After increasing, number of candidates become (140% of 4k), (160% of 5k) & (185% of 6k)
That is, (140x4k)/100, (160x5k)/100 and (185x6k)/100
= 56k/10, 80k/10 and 111k/10

Now, the required new ratio is: 56k/100 : 80k/10 : 111k/10
= 56 : 80 : 111

Hence the answer is option d.

5. The ratio of salary of two persons X and Y is 5:8. If the salary of X increases by 60% and that of Y decreases by 35% then the new ratio of their salaries become 40:27. What is X's salary?

a) Rs.15000 b) Rs.12000 c) Rs.19500 d) data inadequate.

Answer : d) data inadequate

Solution : Ratio of salary of X and Y is 5:8

Let the original salary of X and Y be Rs.5k and Rs.8k respectively.

After increasing 60%, new salary of X = 160% of 5k = 160x5k/100 = 80k/10 ...(1)
After decreasing 35%, new salary of Y = (100-35)% of 8k = 65% of 8k = 52k/10 ...(2)

$$

Given that, new ratio is 40:27
That is, 80k/10 : 52k/10 = 40/27
This does not give the value of k; so that we cannot find X's exact salary.
Hence the answer is data inadequate.