Ratio and Partnership
What is Ratio?
- A ratio is a way to compare two quantities by dividing them to find their simplest form. For
example:
Given the numbers 1000 and 500, if we divide both numbers by their greatest common divisor
(GCD), we
get:
2 : 1
This means the ratio of 1000 to 500 is 2:1, indicating that 1000 is twice as much as 500.
- A ratio is a comparative term. For instance:
If we have the numbers 1000 and 500, we may not immediately understand their relationship.
However,
when we express these numbers as a ratio, like 2:1, it becomes clear that 1000 is twice as much as
500.
Ratios are particularly useful in more complex comparisons.
- A ratio is not an actual value but a representation of the relationship between two quantities. For
example:
If a recipe calls for a ratio of 2 cups of flour to 1 cup of sugar, it means that the amount of
flour is double the amount of sugar, regardless of the actual measurements used.
a : b = 3 : 4 and b : c = 8 : 9
So what will be a : b : c?
a : b | b : c
3 : 4 | 8 : 9
We have 'b' as the common term in both ratios.
To find a common ratio that includes 'a', 'b',
and 'c', we need to make the value of 'b' the same in both ratios.
Currently, 'b' is 4 in the
first ratio and 8 in the second ratio.
To make 'b' the same, we can multiply the first ratio by
2:
2 * (3 : 4) = 6 : 8
Now we have:
a : b | b : c
6 : 8 | 8 : 9
Therefore, the combined ratio a : b : c will be:
a : b : c = 6 : 8 : 9
a : b = 2 : 3, and a : c = 1 : 2
find a : b : c
a : b | a : c
2 : 3 | (1 : 2) * 2
2 : 3 | 2 : 4
a : b : c = 2 : 3 : 4
a : b = 2 : 3, b : c = 4 : 5, c : d = 6 : 7
Find a : b : c : d
x/y = 2/1
Find (x^2 - y^2)/(x^2 + y^2)
- Here we can do it directly:
x/y = 2/1
which means x = 2 and y = 1
so we can directly substitute these values into the equation:
(2^2 - 1^2)/(2^2 + 1^2)
= (4 - 1)/(4 + 1)
= 3/5
x/y = 7/3
find (xy + y^2)/(x^2 - y^2)
- here x is 7 and y = 3
(7*3 + 3^2)/(7^2 - 3^2)
=3/4
(x+y)/(x-y) = 4/1
Find x/y?
- Given:
(x+y)/(x-y) = 4/1
We can write it as:
x + y = 4(x - y)
Now, expand and solve for x and y:
x + y = 4x - 4y
Combine like terms:
x - 4x = -4y - y
-3x = -5y
Divide both sides by -1:
3x = 5y
Divide both sides by y:
x/y = 5/3
A = B + C type questions
- When two things add up to make a third thing. For example, we know:
Expense + Saving = Income
Distance = Speed + Time
Total Marks = Marks Obtained + Marks Lost
Total Cost = Fixed Cost + Variable Cost
Net Profit = Gross Profit - Expenses
Final Price = Original Price + Tax
The income of A and B is 5:4 while their expense is in the ratio of 3:2. If A and B both save 400/-,
find the income of A and B.
- Long form:
The ratio of their incomes is 5:4.
The ratio of their expenses is 3:2.
Both save 400/-, so:
(A's income - 400) / (B's income - 400) = 3 / 2
Let the incomes be 5x and 4x.
Therefore:
(5x - 400) / (4x - 400) = 3 / 2
Cross-multiplying gives:
2(5x - 400) = 3(4x - 400)
10x - 800 = 12x - 1200
12x - 10x = 1200 - 800
2x = 400
x = 200
Therefore, the income of A = 5 * 200 = 1000
The income of B = 4 * 200 = 800
- Short trick:
(A's income - 400) / (B's income - 400) = 3 / 2
When we have the same sign, we subtract, and when we have different signs, we add.
So, 5x and 4x have the same sign, so we first multiply them with the corresponding number on
the RHS:
5x * 2 = 10x
4x * 3 = 12x
Similarly, 400 * 2 = 800 and 400 * 3 = 1200
The difference is:
12x - 10x = 2x and 1200 - 800 = 400
2x = 400
x = 200
Therefore, 5x = 1000 and 4x = 800
The ratio of boys and girls in a college is 5:3. if 50 boys leave the college and 50 girls join the
college, the ratio becomes 9:7. The number of boys in the college is?
-
Start with the given ratios:
(5x - 50) / (3x + 50) = 9 / 7
Check if the terms have the same sign or different signs:
Here, 5x and 3x have different signs.
Multiply 5x and 3x with the corresponding numbers on the RHS:
5x * 7 = 35x
3x * 9 = 27x
Do the same with the changes in numbers (50 boys leave and 50 girls join):
50 * 7 = 350
50 * 9 = 450
Now, find the difference:
35x - 27x = 8x
350 + 450 = 800
Solve for x:
8x = 800
x = 800 / 8
x = 100
The number of boys = 5 * 100 = 500
-
Start with the given ratios:
(5x - 50) / (3x + 50) = 9 / 7
Cross-multiply to simplify:
35x - 350 = 27x + 450
Rearrange to solve for x:
35x - 27x = 800
8x = 800
x = 100
The number of boys = 5x = 500
What must be added to each term of the ratio 2:5 so that it may equal to 5:6?
-
Start with the given ratios:
(2 + x) / (5 + x) = 5 / 6
Cross-multiply to simplify:
6(2 + x) = 5(5 + x)
Expand and solve for x:
12 + 6x = 25 + 5x
6x - 5x = 25 - 12
x = 13
A = B * C
- In this type of question, we deal with the relationship between three factors: quantity (A), price
per unit (B), and the total sale (C).
- It represents a fundamental equation in business and economics, where the total sale (A) is
determined by multiplying the quantity of goods or services sold (B) by their respective unit price
(C).
- For example, if you sell 100 units of a product at $5 each, the total sale (A) would be 100 * $5 =
$500.
The ratio of the salary of each male and female employee in a factory is 3:2. While the ratio of the
number of male and female employees in a factory is 5:4, find the ratio of the total amount
distributed between male and female employees.
The prices of an apple and a mango are in the ratio 3:4. If the cost of 10 apples and 15 mangoes is
Rs. 360, find the cost of 50 apples and 60 mangoes.
- Let's denote the cost of an apple by A and the cost of a mango by M.
- Given the ratio of their prices: A / M = 3 / 4, which implies A = 3k and M = 4k for some
constant k.
- We are given that the cost of 10 apples and 15 mangoes is Rs. 360:
10A + 15M = 360
- Substituting the values of A and M in terms of k:
10(3k) + 15(4k) = 360
30k + 60k = 360
90k = 360
k = 360 / 90
k = 4
- Now, find the cost of 50 apples and 60 mangoes:
A = 3k = 3 * 4 = 12
M = 4k = 4 * 4 = 16
Cost of 50 apples = 50 * 12 = 600
Cost of 60 mangoes = 60 * 16 = 960
Total cost = 600 + 960 = 1560
- Therefore, the cost of 50 apples and 60 mangoes is Rs. 1560.
Partnership
-
Remember: The one who invests more money over a longer period will get better returns.
For example, if A invested Rs. 10,000 for 10 months and B invested Rs. 20,000 for 2 months:
A's total investment = 10,000 * 10 = 100,000 (1 lakh)
B's total investment = 20,000 * 2 = 40,000
So, A will earn more money.
-
Finding the ratio of profit earned by A and B:
Investment ratios: 40,000 : 100,000
Simplified ratio: 2:5
So how do we find this profit share ratio? We calculate it using the money invested and the time
for which it is invested.
-
Concept:
Investment * Time = Profit
In a business, A and B invest Rs. 20,000 and Rs. 30,000 for 8 months and 6 months respectively. Find
the ratio of their profit.
-
A invested Rs. 20,000 for 8 months, so total investment = 20,000 * 8 = 160,000 (160K).
B invested Rs. 30,000 for 6 months, so total investment = 30,000 * 6 = 180,000 (180K).
Now, calculate the ratio of their investments:
160,000 : 180,000
Simplified ratio: 8 : 9
In a business, A and B invest Rs. 30,000 and Rs. 40,000 for 8 months and 9 months respectively. Then
find the share of A and B from the annual profit of Rs. 2500.
A B
30,000 40,000
8 mon 9 mon
--------------------
2,40,000 : 3,60,000
2 : 3
-
So, A has 2 unit shares and B has 3 unit shares from the annual profit of Rs. 2500.
2 units + 3 units = 5 units, which is the total.
5 units = 2500
1 unit = 500
A's share = 2 * 500 = 1000
B's share = 3 * 500 = 1500
In a business, the capital ratio of A, B, and C is 5:6:4 respectively, and the ratio of their time
is 2:3:1. Find the ratio of their profit.
A : B : C
Capital -> 5 : 6 : 4
Time -> 2 : 3 : 1
Profit -> 10 : 18 : 4
-----------------------------
Profit -> 5 : 9 : 2 (answer)
-
To find the profit ratio, multiply the capital ratio by the time ratio for each person.
A's profit ratio: 5 (capital) * 2 (time) = 10
B's profit ratio: 6 (capital) * 3 (time) = 18
C's profit ratio: 4 (capital) * 1 (time) = 4
Therefore, the ratio of their profit is 10:18:4
Simplify the ratio: 10:18:4 = 5:9:2
In a business, the capital ratio of A, B, and C is 3:2:5 respectively, and the ratio of their profit
is 4:3:2. Find the ratio of their time.
A : B : C
Capital -> 3 : 2 : 5
Time -> X : Y : Z
Profit -> 4 : 3 : 2
- To find the time ratio, use the relationship investment * time = profit, so time = profit /
investment.
X = 4 / 3
Y = 3 / 2
Z = 2 / 5
Thus, the time ratio X : Y : Z = 4/3 : 3/2 : 2/5
To simplify, find a common denominator: 4/3 = 40/30, 3/2 = 45/30, 2/5 = 12/30
So, X : Y : Z = 40 : 45 : 12
A, B, and C start a business with an investment of 50,000/-. A invests 4,000/- more than B's
investment. B invests 5,000/- more than C's investment. Find the profit share of A, B, and C from a
profit of 35,000/-.
- If C invested 'x' amount then:
B invested x + 5,000
And A invested x + 5,000 + 4,000 = x + 9,000
Given the total investment is 50,000, then:
x + 9,000 + x + 5,000 + x = 50,000
3x + 14,000 = 50,000
3x = 36,000
x = 12,000
So C's investment = 12,000
B's capital = 12,000 + 5,000 = 17,000
A's capital = 12,000 + 9,000 = 21,000
Now we can determine the profit sharing ratio.
A : B : C
Capital -> 21,000 : 17,000 : 12,000
21 : 17 : 12
- Hence, A has 21 unit shares, B has 17 unit shares, and C has 12 unit shares out of 35,000.
Total share units = 21 + 17 + 12 = 50 units
50 units = 35,000
1 unit = 700
A's share = 21 * 700 = 14,700
B's share = 17 * 700 = 11,900
C's share = 12 * 700 = 8,400
A starts a business with an investment of 30,000/-. After some time, B enters into the business with
an investment of 36,000/-. If their annual profit-sharing ratio is 5:4, then find after how much
time B enters into the business.
- We have an annual profit which means A started his investment from the beginning and continued
for the whole year, which is 12 months.
This means A invested 30,000 for 12 months.
Now, B came after some months which we don't know, and we want to find it.
Let's suppose B invested for x months, and A invested for 12 months. Now, A's total
investment = 30,000 * 12, and B's total investment = 36,000 * x.
Given that the profit-sharing ratio A/B = (30,000 * 12) / (36,000 * x) = 5 / 4,
Simplifying the equation:
(30,000 * 12) / (36,000 * x) = 5 / 4
=> 360,000 / 36,000x = 5 / 4
=> 10 / x = 5 / 4
=> 10 * 4 = 5 * x
=> 40 = 5x
=> x = 8
Hence, B invested for 8 months. Therefore, 12 - 8 = 4.
So, after 4 months, B started his investment.
A and B invest 50,000 and 40,000 in a business. After 4 months, A withdraws 10,000 and after 2
months of A's withdrawal, B invests 10,000 in the business. Then find the ratio of their profits.
- If it is not given for how many months or years they have invested, we assume it is for 1 year.
- A invested 50,000 for 4 months, so total investment till 4 months = 50,000 * 4 = 200,000.
Then A withdraws 10,000, so for the next 8 months, he invested 40,000. Total for 8 months =
40,000 * 8 = 320,000.
So, for the whole year, A's total investment = 200,000 + 320,000 = 520,000.
- B invested 40,000 for 6 months, so total investment till 6 months = 40,000 * 6 = 240,000.
Then B adds 10,000, so for the next 6 months, he invested 50,000. Total for 6 months =
50,000 * 6 = 300,000.
So, for the whole year, B's total investment = 240,000 + 300,000 = 540,000.
- Now the profit share ratio = A's total investment : B's total investment
Profit share ratio = 520,000 : 540,000
To simplify the ratio, we divide both by the GCD of 520,000 and 540,000, which is 20,000.
Profit share ratio = (520,000 / 20,000) : (540,000 / 20,000) = 26 : 27
A and B invests 30,000 and 40,000 in a business, after 3 month A invests 10,000 more and after 3
months of A's investment B withdraw 10,000 from this business then find the ratio of their profits.