RATIO AND PROPORTION

RATIO

  Ratio means the number of times one quantity contains another quantity of the same kind. The comparison is made by considering what part or multiple the first quantity of the second.
       Thus the ratio between Rs 50 and Rs 200 can be possible, but not between Rs 50 and 200 marks.
       The ratio between one quantity to another is measured by  a : b or a/b
^{Ex: 8:9   or   5:7 etc.
       The two quantities in the ratio are called its terms. The first is called the antecedent and the second term is called consequent.}
 

^{Types of Ratios:}
 

^{1. Duplicate ratio: The ratio of the squares of the two numbers.
     Ex: 9 : 16 is the duplicate ratio of 3 : 4.}
 

^{2. Triplicate Ratio: The ratio of the cubes of the two numbers.
     Ex: 27 : 64 is the triplicate ratio of 3 : 4.}

<sup>3. Sub-duplicate Ratio: The ratio between the square roots of the two numbers.
      Ex: 4 : 5 is the sub-duplicate ratio of 16 : 25.
 
4. Sub-triplicate Ratio: The ratio between the cube roots of the two numbers.
      Ex: 4 : 5 is the sub-triplicate ratio of 64 : 125.
 
5.Inverse ratio: If the two terms in the ratio interchange their places, then the new ratio is inverse ratio of the first.
      Ex: 9 :5 is the inverse ratio of 5 : 9.
 
6. Compound ratio: The ratio of the product of the first terms to that of the second terms of two or more ratios.
      Ex: The compound ratio of</sup>  3/4, 5/7, 4/5, 4/5 is 9/35
 



^{PROPORTION 
If two ratios are equal, then they make a proportion.
 Thus}  4/5 = 8/10 or 4:5 = 8:10
^{Each term of the ratios 4/5 and 8/10  is called proportional.
The middle terms 5 and 8 are called means and the end terms 4 and 10 are called extremes.}

Product of Means = Product of Extremes

^{Continued Proportion: In the proportion  8/12= 12/8  8, 12, 18 are in the continued proportion.}
 

^{Fourth proportion: If a : b = c : x, then x is called fourth proportion of a,b and  c.
There fore fourth proportion of  a, b, c  =} b x c/a
 

^{Third proportion: If a : b = b : x, then x is called third proportion of a and b.
Therefore third proportion of a, b =  b^2/a}
 

^{Second or mean proportion: If a : x = x : b , then x is called second or mean proportion of a and b.
Therefore mean proportion of a and b =  root(ab)}

EXAMPLES

^{Example 1: Find out the two quantities whose difference is 30 and the ratio between them is 5/11.
Sol: The difference of quantities, which are in the ratio 5:11, is 6. To make the difference 30, we should Multiply them by 5.
        Therefore  5:11 = 5x5 : 11x5 = 25 : 55}
 

^{Example 2: A factory employs skilled workers, unskilled workers and clerks in the ratio 8:5:1 and the wages of a skilled worker, an unskilled worker and a clerk are in the ratio 5:2:3 when 20 unskilled workers are employed the total daily wages fall amount to Rs. 318. Find out the daily wages paid to each category of employees.
Sol: Number of skilled worker: unskilled worker: clerks = 8:5:1 and the ratio of their respective Wages = 5:2:3
       Hence the amount will be paid in the ratio 8 × 5 : 5 × 2 : 3 × 1 = 40 : 10:3          
       Hence total amount distributed among unskilled workers
                                   318  x 10 / (40+10 +3)} = Rs 60.

^{But the number of unskilled workers is 20, so the daily wages of unskilled worker
                                                      60/20 = rs.30
The wages of a skilled worker, an unskilled worker and a clerk are in the ratio = 5:2:3
      Multiplying the ratio by (5/2) and (3/2)we get = 7.50 : 3 : 4.50
      So, if an unskilled worker gets Rs.3 a day then a skilled worker gets Rs. 7.50 per day a clerks Rs. 4.50 a day
 
Example 3: Two numbers are in the ratio of 11:13. If 12 be subtracted from each, the remainders are in the ratio of 7:9 Find out the numbers.
Sol: Since the numbers are in the ratio of 11:13. Let the numbers be 11x and 13x. Now if 2 is subtracted from each, the numbers become (11x -12) and (13x-12).  As they are in the ratio of 7:9              (11x-12): (13x-12):: 7: 9
                (11x – 12) 9 = (13x – 12) 7
                99x – 108 = 91x – 84
                9x = 24 or x = 3
  Therefore the numbers are 11 x 3 = 33 and 13 x 3 = 39}

Example 4:  In what ratio the two kinds of tea must be mixed together one at Rs. 48 per kg. and another at Rs. 32 per kg. So that the mixture may cost Rs. 36 per kg. ? Sol:  Ratio of inferior quality to superior quality=(rate of superior - rate of mix)/(rate of mix- rate of inferior) = (48-36)/(36-32) =12/4 =3:1

^{Example 5:  If Rs. 279 were distributed among Ram, Mohan and Sohan in the ratio of 15:10:6 respectively, then how many rupees did Mohan obtain?
Sol: Ratio in which Ram, Mohan and Sohan got  = 15 :10 : 6
                        Sum of ratios  = 15 + 10 + 6 = 31
                        Share of Mohan} = (10 x 279)/ 31
                                                         ^{= Rs. 90
 
Example 6:  A bag contains of one rupee, 50 paise and 25 paise coins. If these coins are in the ratio of 2:3:10, and the total amount of coins is Rs288, find the number of 25 paise coins in the bag.
Sol:   Ratio of one rupee, 50 paise and 25 paise coins
                                                                 = 2:3:10
                        Ratio of their values  = 8:6:10 = 4:3:5
 And sum of the ratios of their values = 4 + 3 + 5 = 12
                     Value of 25 paise coins (5x288)/12 = Rs. 120
                      No. of 25 paise coins   = 120 × 4 = 480}                

Percantage change Or Profit-loss

1. The price of petrol is increased by 25%. By what percent the consumption be reduced to make the expenditure remain the same?

a. 25%             b. 33.33%                    c. 20%             d. None

2. If the length of a rectangle is increased by 33.33%, by what percentage should the breadth be reduced to make the area same?

a. 20%             b. 33.33%                    c. 25%             d. None

Have the below list::::::::::::::::

  

10%                            0.1                               1/10

12.5%                         0.125                           1/8

16.66%                       0.1666                         1/6

20%                            0.2                               1/5

25%                            0.25                             1/4

30%                            0.3                               3/10

33.33%                       0.3333                         1/3

40%                            0.4                               2/5

50%                            0.5                               1/2

60%                            0.6                               3/5

62.5%                         0.625                           5/8

66.66%                       0.6666                         2/3

70%                            0.7                               7/10

75%                            0.75                             3/4

80%                            0.8                               4/5

83.33%                       0.8333                         5/6

90%                            0.9                               9/10

100%                          1.0                                1

Short trick for above problems:

in first problem,he want to bring back the original value,so need to decrease the value.

price increased,so became–25%

in the above list,see the next decreased value from 25% IS 20% so yopur answer is 20%

in second problem

increased value——–33.33%

next decreased value–30%,SO ANSWER IS 30%

Percentage Change:

A change can be of two types – an increase or a decrease.                    

When a value is changed from initial value to a final value,

% change = (Difference between initial and final value/initial value) X 100

Eg: If 20 changes to 40, what is the % increase?

Soln: % increase = (40-20)/20 X 100 = 100%.

Note:

1. If a value is doubled the percentage increase is 100.

2. If a value is tripled, the percentage change is 200 and so on.

Percentage Difference:

% Difference = (Difference between values/value compared with) X 100.

Eg: By what percent is 40 more than 30?

Soln: % difference = (40-30)/30 X 100 = 33.33%

(Here 40 is compared with 30. So 30 is taken as denominator)

Eg: By what % is 60 more than 30?

Soln: % difference = (60-30)/30 X 100 = 100%.

(Here is 60 is compared with 30.)

Hint: To calculate percentage difference the value that occurs after the word “than” in the question can directly be used as the denominator in the formula.

Understanding Profit and Loss:

So, by now we came to know that if CP is increased and sold it would result in profit and vice versa.

Also whatever increase is applied to CP, that increase itself is the profit.

For Rs. 10 profit, CP is to be increased by RS. 10 and the increased price becomes SP.

For 10% profit, CP is to be increased by 10% and it is the SP.

(From previous chapter we know that any value increased by 10% becomes 1.1 times.)

So, for 10% profit, CP increased by 10% => 1.1CP = SP.

·        SP = 1.1CP => SP/CP = 1.1 => 10% profit

·        SP = 1.07CP => SP/CP = 1.07 => 7% profit

·        SP = 1.545CP => SP/CP = 1.545 => 54.5% profit and so on.

Similarly,

·        SP = 0.9CP => SP/CP = 0.9 => 10% loss (Since 10% decrease)

·        SP = 0.76CP => SP/CP = 0.76 => 24% loss and so on.

So, to calculate profit % or loss %, it is enough for us to find the ratio of SP to CP.

Note:

1.      If SP/CP > 1, it indicates profit.

2.      If SP/CP < 1, it indicates loss.

Multiple Profits or losses:

A trader may sometimes have multiple profits or losses simultaneously. This is equivalent to having multiple changes and so all individual changes are to be multiplied to get the overall effect.

Relationship among CP, SP and MP:

A trader adds his profit to the investment and sells it at that increased price.

Also he allows a discount on Marked Price and sells at the discounted price.

So, we can say that,

SP = CP + Profit. (CP applied with profit is SP)

SP = MP – Discount. (MP applied with discount is SP)

Q: A trader uses 1 kg weight for 800 gm and increases the price by 20%. Find his profit/loss %.

Soln: 1 kg weight for 800 gm => loss (decrease) => 800/1000 = 0.8

20% increase in price => profit (increase) => 1.2

So, net effect = (0.8) X (1.2) = 0.96 => 4% loss.

2.      A milk vendor mixes water to milk such that he gains 25%. Find the percentage of water in the mixture.

Soln: To gain 25%, the volume has to be increased by 25%.

So, for 1 lt of milk, 0.25 lt of water is added => total volume = 1.25 lt

% of water = 0.25 / 1.25 X 100 = 20%.

Percentages trick

Percentages trick

Find 7 % of 300. Sound Difficult?

Percents: First of all you need to understand the word “Percent.” The first part is PER , as in 10 tricks per list verse page. PER = FOR EACH. The second part of the word is CENT, as in 100. 

Like Century = 100 years. 100 CENTS in 1 dollar… etc. Ok… so PERCENT = For Each 100.

So, it follows that 7 PERCENT of 100, is 7. (7 for each hundred, of only 1 hundred).

8 % of 100 = 8. 35.73% of 100 = 35.73

But how is that useful??

METHOD 1

Back to the 7% of 300 question. 7% of the first hundred is 7. 7% of 2nd hundred is also 7, and 7% of the 3rd hundred is also 7. So 7+7+7 = 21.

If 8 % of 100 is 8, it follows that 8% of 50 is half of 8 , or 4.

Break down every number that’s asked into questions of 100, if the number is less then 100, then move the decimal point accordingly.

EXAMPLES:

8%200 = ? 8 + 8 = 16.

8%250 = ? 8 + 8 + 4 = 20.

8%25 = 2.0 (Moving the decimal back).

15%300 = 15+15+15 =45.

15%350 = 15+15+15+7.5 = 52.5

Also it’s useful to know that you can always flip percents, like 3% of 100 is the same as 100% of 3.

35% of 8 is the same as 8% of 35

METHOD 2

In other words, divide the 'percentage number'

15%500 =  (10%+5% =15%) now 10%500=50 and (10/2 = 5) so 50/2=25  now add both 50+25 = 75 is ur answer

20%490= (10% + 10%).. 49+49 = 98

27.5 %444 = (25% +2.5%).. you just need to shift '.' by one left.. 111+11.1 = 122.1

33% 312= (30% + 3%) or (10+10+10+ 1+1+1) or here its easy to find 3times of 312 = 936 

so that 936 will be 300% and 30% will be 93.6 and 3% will be 9.36

For answer add 93.6 and 9.36 =   104.66