Ratio, Proportion & Partnership: (part 2 )Exercise Problem & solved examples

Question: 10 Given (P + 7Q)/4P = 19/20, what is the ratio of Q:P?
Option: (a) 1:2 (b) 1:3 (c) 2:4 (d) 2:5 (e) 2:7

Explore: (P + 7Q)/4P = 19/20
=> 20P + 140Q = 76P
=> 140Q = 56P
=> P/Q = 140/56 = 5/2
=> P:Q = 5:2
:. Q:P = 2:5

Answer: (d)

Question: 11 Nine years ago the age of P and Q| were in the ratio of 2:3. After 7 years, the ratio of their age will be 3:4. What is the present age of P?

Explore: Let 9 years ago the age of P and Q were 2x & 3x respectively.
:. Present age of P = 2x + 9 
Present age of Q = 3x + 9
According to the question,
(2x +9+7):(3x+9+7) = 3:4
=> (2x+16)/(3x+16) = 3/4
=> 4(2x+16) = 3(3x +16)
=> 8x + 64 = 9x +48
=> 9x-8x = 64 - 48
=> x = 16
:. Present age of P = 2 x 16 + 9 = 41
Answer: 41

Question: 12 Age of three persons are now in the proportion 2:3:4 and in 5 years from now, the proportion will be 5:7:8. What is the present age of the youngest person?
Option: (a) 30 (b) 25 (c) 20 (d) 15 (e) None

Explore: The ratio of the age of the three persons is 2:3:4. So, their age can be 2X, 3X, 4X, where X is any number.
After 5 years their age ratio is given by 5:7:8. So, their age may be 5Y, 7Y & 8Y where Y is any other number.
So, we can write
2X+5 = 5Y .......(1)
3X+5 = 7Y .......(2)

(2) - (1) gives X = 2Y
Or, Y = 1/2
Replacing this value in (1)
2X +5 = (5/2)X
Or, 5 = ((5/2) - 2)X = (1/2)X
:. X = 10
:. Required age 2X = 2x10 = 20
Answer: (c)

Question: 13 A fruit salad mixture consists of apples, peaches and grapes in the ratio 6:5:2 respectively, by weights. If 39 pounds of the mixture is prepared, the mixture includes how many more pounds of apples than grapes?

Explore: In the mixture of 39 pounds,
ratio of apples, peaches & grapes = 6:5:2
:. Weight of apples = 39 x (6/(6+5+2)) = 39 x (6/13) = 18 Pounds
Weight of peaches = 39 x (5/13) = 15 Pounds
Weight of grapes = 39 x (2/13) = 6 Pounds
:. Weight of apples is (18 - 6) or 12 pounds more than grapes.
Answer: 12 pounds

Question: 14 Jahirul and Jalil agree to form a partnership. The partnership agreement requires that Jahirul invest $7,000 less than one half of what Jalil invests. If the total investment is $ 1,25,000 how much does Jalil invest?

Explore: Let, Jalil invest $X
:. Jahirul invests $ (X/2) - 7000
According to the question,
X+(X/2) - 7000 = 125000
=> (2X+X)/2 = 125000+7000
=> 3X/2 = 132000
:. X = (132000 x 2)/3
=> X = 88000
Answer: Jalil invests $ 88000

Question: 15 Arif, Babu and Salm started a business jointly with a total amount of $28000. Arif paid $ 4500 more than Babu and Babu paid $7000 less than Salam. If the company made a profit of $5600, how much profit should Babu receive?

Explore: Let Babu paid $X
:. Arif paid $(X+4500)
Salam paid $(X+7000)
According to the question,
X+X+4500+X+7000 = 2800
=> 3X + 11500 = 28000
=> 3X = 28000 - 11500
=> X = (16500/3) = 5500
:. Babu paid $ 5500
Arif paid $ (5500+4500) = $10000
Salam paid $ (5500 + 7000) = $12500
They will share profit according to the money they paid.
:. Ratio of money paid by Babu, Arif and Salam
= 5500:10000:12500
= 55:100:125
= 11:20:25
Some of the ratio = 11+20+25 = 56
:. Babu received from the profit of $5600
= $5600 x (11/56)
= $1100
Answer: $1100

Question: 16 Arif, Babu and Salam started a business jointly with a total amount of $280. Arif paid $45 more than Babu and Babu paid $70 less than Salam. If the company made a profit of $56, how much profit should Babu receive?
Option: (a) 22 (b) 20 (c) 25 (d) 27 (e) None

Explore: Let, Babu's investment be $B
:. Arif's investment is $(B+45)
And Salam's investment $ (B+70)
According to the problem,
b+(b+45)+(b+70) = 280
=> 3B = 280 -115 = 165
:. B = 55
Since the profit is shared in proportion of capital investment, so,
Babu's profit = ((Babu's capital)/(Total capital)) x Total profit = ((55)/(250)) x 56 = $11
Answer: (e)

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Ratio, Proportion & Partnership: Exercise Problem & solved examples

Question: 1 The ratio of 1/5 to 2/7 is:
Option: (a) 3:5 (b) 5:7 (c) 7:9 (d) 7:10 (e) 3:7

Explore: The ratio of 1/6 to 2/7
= 1/5:2/7
= (1/5)/()2/7
= 1/5 x 7/2
= 7/10
= 7: 10
Answer: (d)

Question: 2 The ratio of 1/4 to 3/5 is
Option: (a) 1 to 3 (b) 3 to 20 (c) 5 to 12 (d) 3 to 4 (e) 5 to 5

Explore: 1/4 to 3/5 = 1/4:3/5 = (1/4)/(3/5) = 5/12
Answer: (c)

Question: 3 A:B = 4:5, A:C = 10:9, then A:B:C = ?
Option: (a) 4:5:9 (b) 4:5:10 (c) 8:9:10 (d) 8:9:10 (e) 20:25:18

Explore: A:B = 4:5
=> B:A = 5:4
=> B:A = 5:5 x 4:5 = 25:20
A:C = 10:9
=> A:C = 10x2:9x2 = 20:18
:. B:A:C = 25:20:18
:. A:B:C = 20:25:18
Answer: (e)

Question: 4 If A:B = 1:2, B:C = 4:3, & A+B+C = 450. What is the value of B?
Explore: A:B = 1:2 = 1x2:2x2 = 2:4
B:C = 4:3
:.  A:B:C = 2:4:3
Sum of the numbers = 2+4+3 = 9
:. Value of B = 450x(4/9) = 200
Answer: 200

Question: 5 Let, X:Y = 3:4 and X:Z = 6:5, then Z:Y =?
Option: (a) 5:3 (b) 6:7 (c) 4:2 (d) 5:4 (e) 5:8

Explore: X:Y = 3:4
=> Y:X = 4:3 => Y:X = 4x2:2x3 = 8:6
Again, X:Z = 6:5
:. Y:X:Z =8:6:5
:. Z:Y = 5:8
Answer: (e)

Question: 6 If X:Y = Y:Z = 1.5 and Z = 2, what is the value of X?
Option: (a) 3 (b) 4 (c) 4.5 (d) 2.5 (e) 5

Explore: Y:Z = 1.5 or Y/Z = 1.5
Or, Y = 1.5Z = 1.5x2 = 3
Now, Z:Y = 1.5
:. X = 1.5Y = (3/2)3 = 9/2 = 4.5
Answer: (c)

Question: 7 If X is 2/5 of Y and Y is 5/7 of Z, what is the ratio of Z:X?
Explore: According to the question,
X = 2Y/5 & Y = 5Z/7
=> X = 2/5 x 5Z/7
=> X = 2Z/7
=>Z/X = 7/2 = 7:2
Answer: 7:2

Question: 8 In a school, the ratio of boys to girls is 3 to 7. If there are 150 boys and girls in the school, how many boys are there?
Option: (a) 45 (b) 75 (c) 90 (d) 105

Explore: The ratio of boys to girls = 3:7
Sum of the ratio = 3 + 7 = 10
No. of boys = 150 x 3/10 = 45
Answer: (45)

Question: 9 The ratio of girls to boys in a class is 3:4. The number of students in the class could be any of the following except:
Option: (a) 35 (b) 28 (c)48 (d) 42

Explore: Since ratio of girls to boys is 3:4 and no. of student can't be fraction and must be divisible by 7(3+4), so, 35, 28 & 42 are divisible by 7.
But 48 is not divisible by 7.

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Ratio, Proportion & Partnership: Ration, Rule, Proportion, Rate, Partnership

Ratio: A ratio is a comparison of two quantities in the same units. We generally separate the two quantities in the ratio with a colon (:). Suppose, we want to write the ratio of 8 and 12. We can write this as 8 : 12 or as a faction 8/12
First term of a ratio is called antecedent, while the second term is known as consequent. Here, 8 is antecedent & 12 is consequent.

Rule: Multiplication or division of each term of a ratio by a same non-zero number does not affect the ratio.
Ratio of 8 & 12 = 8:12 = 8/12 = 4/6 = 2/3 = 10/15 = 14/21
i.e 8:12 = 4:6 = 2:3 = 10:15 = 14:21
3:4 = 2x3:4x2 = 6:8
10:5 = 10/5:5/5 = 12:1

Proportion: A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. Simply, the equality of two ratios is called proportion.
3/4 =6/8 or 3:4 = 6:8 is an example of proportion.
From 2/3 = 6/9
We can write, 2x9 = 3x6 [Cross-multiplication]

Rate: A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/hr. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator.

Ratio is a relationship: a number of people to a number of people; an amount of money to an amount of money.
<> A:B = 2:3; B:C = 3:4
:. A:B:C = 2:3:4
<> A:B = 2:3; B:C = 4:5
A:B = 2:3 = 2x4:3x4 = 8:12
B:C = 4:5 = 4x3: 5x3 = 12:15
:. A:B:C = 8:12:15

Partnership: Whenever two or more people involve in a business, invest their money and share profit according to the ratio of the ratio of their investment, partnership arises.

For example:
* A, B & C form a partnership & invest $60,000 $80,000 & $100,000 respectively. After one year, they receive $24,000 as profit. What will be the profit of each?

Explore: Ratio of investment
= 60,000 : 80,000 : 100,000
= 60 : 80 : 100
= 6 : 8 : 10
= 3 : 4 : 5
Sum of the numbers = 3+4+5 = 12
:. Profit of A = $24,000 x (3/12) = $6,000
Profit of A = $24,000 x (4/12) = $8,000
Profit of A = $24,000 x (5/12) = $10,000

*Divide 70 chocolates among girls & boys, where ratio of girls to boys is 3:4.

Explore: Ratio of girls to boys = 3:4
Sum of the terms = 3+4 = 7
:. Girls will get = 70 x (3/7) = 30 Chocolates
Boys will get = 70 x (4/70) =40 Chocolates

* 3:X = 2:6, X = ?

Explore: 3:X = 2:6
=> 3/X = 2/6
=> 3x6 = 2xX
=> X = 18/2 = 9
Answer: X = 9

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