Formula for ratio

A : B => a / b

Formula for proportion 

a : b :: c : d => a/b = c /d

Properties of ratio and proportion

(a) Invertendo represented by a/b = c/d => b/a = d/c

Proof:

a : b :: c : d

⟹ ab�� = cd��

⟹ ba�� = dc��

⟹ b : a :: d : c

(b) Alternendo represented by a/b = c/d => a/c = b/d

(c ) Componendo represented by a/b = c/d => (a + b)/b = (c+d)/d

(d) Dividendo represented by a/b = c/d => (a – b)/ b = (c – d)/ d

• In the case of proportion, when two ratios are equal, it is proportional to each other. It is presented as a product of extremes equal to the product of means.

a: b:: c : d, where a and d are the extremes and b and c are the means.

• When a, b, c and d are placed in continued proportion, it is presented as a:b = b:c = c:d
• For example, in the equation a:b = b:c, b is the mean proportion. Also, b2 = ac.
• In an equation a:b = b:c, c is the considered third proportion to the numbers a and b
• However, if there is a ratio and proportion a:b = c:d, then d is the fourth proportional to the numbers a, b and c.

Continued proportion

• Three quantities a, b and c are said to be in continues proportion if a:b::b:c . If a:b:b:c , then c is called the third proportional of a and b.

• If a : b = c : d, then:
  • d is called the fourth proportional to a, b, c.
  • c is called the third proportional to a and b.
  • The mean proportional between a and b is √(ab).

Example 1: Divide Rs. 60 in the ratio 1:2 between Mike and John.

Solution: Let Mike’s part be x.

Then John’s part is 2x.

Thus, x+2x = 60

=> 3x = 60

=> x = (60/3)

=> x = 20.

Therefore, Mike’s part = x = Rs. 20

John’s part = 3x = Rs. (2*20) = Rs. 40

Example 2: If there are Rs. 495 in a bag in denominations of one-rupee, 50-paisa and 25-paisa coins which are in the ratio 1:8:16. How many 50 paisa coins are there in bag?

Solution: Assume, you have x numbers of one rupee coin. Now coins are in the ratio 1:8:16. This means that if we have x number of one rupee coins, we have 8x number of 50 paisa coins and 16x number of 25 paisa coins. Here order in which ratios are mentioned in the question is very important. In this case order is one rupee, 50 paisa and 25 paisa and ratio is 1:8:16.Thus,

Number of 50-paisa coins = 8x

Number of 25- paisa coins = 16x

Now,

Total money in the bag = Rs. 495

• x+ (8x/2) + (16x/4) = 495

(50 paisa coins divided by 2 to convert into rupee and 25 paisa coins divided by 4 to convert into rupee)

• 9x = 495
• x = 495/9
• x= 55

Thus, number of 50 paisa coins = 55*8 = 440

Example 3: Three Jars contain alcohol to water in the ratios 3:5, 1:3 and 1:1. If all the three solutions are mixed, what will be the ratio of alcohol to water in final solution?

Solution: Here we are not given the quantities of the solution in three jars. Only the ratio of alcohol to water is given. If the ratio of quantity of solution would have been there, we could determine the ration of alcohol to water in final solution. Hence, the answer here will be cannot be determined.

Example 4: A lump of two metals weighing 18gm is worth Rs. 87. If their weights are interchanged, it would be worth Rs. 78.6. If the price of one metal is Rs. 6.7 per gram, find the weight of the other metal in the mixture.

Solution: Let the weight of first metal be x. Then the weight for other metal would be (18-x).

Also, let the price of other metal be Rs. K per gram. We know that price for the first metal is Rs. 6.7. Thus

Now, Price for first metal = Rs. 6.7 per gram

Price for second metal = Rs. K per gram

Originally, Weight of first metal = x gm and weight of second metal = (18-x) gm

Also, originally worth of the lump = Rs. 87

Therefore, 6.7x + K (18-x) = 87

When the weights are interchanged, Weight of the first metal = (18-x) gm and Weight of second metal = x gm

Therefore, 6.7(18-x) + Kx = 78.6

Adding the two equations above:

6.7x + 18K – Kx + (6.7*18) -6.7x +Kx = 87+ 78.6

=>18K + 120.6 = 165.6

=>18k = 45

=>K= 45/18 = 5/2

Substituting value of K= 5/2 in equation 1

6.7x + (5/2 * (18 – X)) = 87

=>       x = 10

Thus, weight of other metal in the mixture = (18-10) gm = 8 gm

Example 5: A bottle of whisky contains ¾ of whisky and the rest is water. How much of the mixture must be taken away and substituted by equal quantity of water so as to have half whisky and half water?

Solution:  We will use the formula below for answering this. This is a standard formula for mixtures and very useful when questions concerning mixture and ratio comes in.

F = I (1- X/V)

F= Fraction of whisky in final mixture = 1/2

I = Fraction of whisky in Initial mixture = 3/4

X= Quantity of mixture replaced = ??

V= Total Volume of mixture

Substituting the values in above formula

=>½=3/4 (1-X/V)

=>(1-X/V) = (1/2)*(4/3)

=>(1-X/V) = 2/3

=>X/V = 1/3

So, the answer is Quantity of mixture replaced would be 1/3 of the original mixture.

Example 6: A man covered a total distance of 1,000 km in 16 hr, partly in a taxi at 36 km/hr and partly in a bus at 80 km/hr. The distance covered by the bus is approx.?

a.) 770 km          b.)640 km            c.) 720 km           d.) 680 km       e.) None of the above

Solution: This is a classic example of how ratios questions comes combined with the concept of speed, time and distance. Now let’s look at how we can solve this particular question.

We know,   Speed = distance / time.

Now, Let the distance covered by bus be x km

Then, distance covered by taxi = (1000-x) km

Also, Time taken by the bus to cover x km = Distance/Speed = x km / 80 km/hr

= x/80 hr.

Similarly, time taken by the taxi to cover (1000-x) km = (1000-x)/36 hr.

We know as per question that total time taken is 16 hr.

So,   (x/80) + ((1000-x)/36) = 16

=>X =~ 770

Thus, answer would be, Distance covered by the bus = 770 km

Example 7: In a mixture of 35 L, the ratio of milk to water is 4:1. If 7 L water is added to the mixture, the ratio of milk to water changes to a new ratio. If we want ratio of milk and water to change back to the original value, how much milk is to be added now?

Solution:  Initial Ratio of milk: water = 4:1

Let the quantity of water be x. Then, quantity of milk would be 4x.

As per question,

x+4x = 35

=>x = 7

In original mixture,

Quantity of water = 7 L

Quantity of milk = 28 L

After 7 L of water is added to the mixture,

Quantity of water = 14 L

Quantity of milk = 28 L

New ratio of milk: water = (28:14) = (2:1)

Original ratio = 4:1

Let the quantity of milk to be added = x L

Thus,  New quantity of milk / New quantity of water = 4/1

=>(28+x) / 14 = 4/1

=>28+x = 56

=>x = 56-28 = 28

Hence, quantity of milk to be added = 28 L

Example 8: The ratio of a two-digit natural number to a number formed by reversing its digits is 4 : 7. Which of the following is the sum of all the numbers of all such pairs?
a. 99            b. 198                 c. 330              d. 132

Solution: Let the two digit number be 10a + b and the number formed by reversing its digits be 10b + a.

(10a + b) / (10b + a) = 4/7

70a + 7b = 40b + 4a

66a = 33b

Therefore,

a/b = 1/2

So, let us list down all possible values for a and b.

a             b             Number               Reversed Number

1             2             12                               21

2             4             24                               42

3             6             36                               63

4             8             48                                84

 

Hence, the sum of all the numbers would be,

12 + 21 + 24 + 42 + 36 + 63 + 48 + 84 = 330.

Answer choice (c)

Example 9: Rs.432 is divided among three workers A, B and C such that 8 times A’s share is equal to 12 times B’s share which is equal to 6 times C’s share. How much did A get?

Solution:

As per question,

8A = 12B = 6C

Expressing B’s and C’s share in terms of A’s share

B = (8/12) A = (2/3) A        (Since 8A = 12 B)

C = (8/6) A = (4/3) A            (Since 8A = 6C)

Now  A+B+C = 432

=>A + (2/3)A + (4/3)A = 432

=>3A = 432

=>A = 144

Thus, A’s share is Rs. 144

Example 10:  Three travellers are sitting around a fire, and are about to eat a meal. One of them has 5 small loaves of bread, the second has 3 small loaves of bread. The third has no food, but has 8 coins. He offers to pay for some bread. They agree to share the 8 loaves equally among the three travellers, and the third traveller will pay
8 coins for his share of the 8 loaves. All loaves were of the same size. The second traveller (who had 3 loaves) suggests that he will be paid 3 coins, and that the first traveller be paid 5 coins. The first traveller says that he should get more than 5 coins. How much should the first traveller get?

a.) 5

b.) 7

c.) 1

d.) None of these

Solution:  Each traveler had 8/3 loaves after the distribution of the loaves.

First traveler has given (5- 8/3) = 7/3 loaves to the third one.

Second traveler had given (3- 8/3) = 1/3 loaves to the third one.

Since the first traveler had given 7 times the loaves of the second traveler , the coin should be distributed in the ratio of 7: 1

Hence, first traveler should get 7 coins and second traveler should get 1 coin.

Answer Choice: B