Ratio and Proportion  Atreya Roy

Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.
When we have two items and we need to relate them with respect to amount or quantity or any other measure, we use Ratio. Ratio is nothing but a relational number that speaks for the two items which we are considering. But ratio is the relation between two quantities only when they are in the same units. You cant compare kg with litres.
If we say $\frac{a}{b} = \frac{2}{3}$ that means $a,b$ are in a ratio of $2:3$ which means for 2 units of $a$, there are 3 units of $b$
Example : The price of two shirts S1 and S2 are 100 and 160 respectively. What is the ratio of the prices ?
Solution : $\frac{S1}{S2} = \frac{100}{160}$. Simply calculate them to their simplest form. = $\frac{5}{8}$
So the prices are in a ratio = $5:8$
Compounded ratio :
When two ratios are multiplied then the ratio obtained is called compounded ratio.
$a:b$, $c:d$ after compounding will be $(\frac{a}{b})\times (\frac{c}{d}) = \frac{ac}{bd}$
Proportion:
When two ratios are equal, then the terms in the ratios are said to be in proportion
$a:b::c:d = \frac{a}{b} = \frac{c}{d}$
or, $ad = bc$
Question:
What should be added to each of the numbers 17, 35, 39, 79 so that the resulting number should be in proportion?
Solution : From the formula of proportion we can write : $\frac{(17+x)}{(35+x)} = \frac{(39+x)}{(79+x)}$
Cross multiplying, we get –
$1343+96x+x^2 = 1365+74x+x^2$
Or, $22x=22$, or $x=1$
Putting $x=2$, we get : $\frac{18}{36} = \frac{40}{80} = 1:2$
Direct  Indirect Proportions
Many a times we come across the term “proportional to”. Prices are proportional to quantities or kind of material. This means the measure is varying with respect to another attribute of the item.
Suppose we say the price of chocolates are proportional to the quantity you buy them. For 10 chocolates the price is Rs 20. For 15, its Rs 28. So with increase in quantity, the price decreases. Hence the price depends on various attributes/variables.
When we say, $A$ is directly proportional to $B$, we mean:
$A= k \times B$, where $k$ is the constant of proportionality.
When we say, $A$ is inversely proportional to $B$, We mean :
$A = \frac{k}{B}$ or $AB=k$, where $k$ is the constant of proportionality.
Example :
The cost of carrying luggage is proportional to square of the Weight of the luggage. When the weight is 4Kg, the luggage carry cost is Rs 160. What is the cost when the luggage is 6 kg?
Solution : $cost = K \times weight^2$
$160 = k \times 4^2$ Or, $k= 10$
So, when weight=6kg,
$cost= k \times 6^2 = 10 \times 36 = 360$
Problems on Ages
Ten years ago, P was half of RON's age. If the ratio of their present ages is 3:4, what will be the total of their present ages?
Solution :
Let present age of P and Ron be 3x and 4x respectively.
Ten years ago, P was half of RON's age⇒(3x−10)= (1/2)*(4x−10)
⇒6x−20=4x−10
⇒2x=10
⇒ x=5
⇒(3x−10)=12(4x−10)⇒6x−20=4x−10⇒2x=10⇒x=5
Example :
Father is aged three times more than his son Ram. After 8 years, he would be two and a half times of Ram's age. After further 8 years, how many times would he be of Ram's age?
Solution :
Assume that Ram's present age =x
Then, father's present age =3x+x=4x
After 88 years, father's age =2(1/2) times of Rams' age
⇒(4x+8 )= (5/2)*(x+8 )
⇒8x+16=5x+40
⇒3x=40−16=24
⇒x=24/3=8
After further 8 years,
Ram's age =x+8+8=8+8+8=24
Father's age =4x+8+8=4×8+8+8=48
Father's age/Ram's age =48/24=2
Partnership
Many a time a new business is set up by a bunch of partners. This may on various terms. Some invest their money on the business. So all these leads to the division of profit that the business incurs after a certain amount of time. We will have to deal with problems which consider 23 partners in a business. Generally the ratio of profit depends on the investment amount in the business and the time for which it was kept into the business i.e., the period of investment.
Suppose X and Y are two partners in a business. They invested amount A and B for time C and D months.
So profit ratio will be calculated as follows :
Let the Total Profit be = P
X’s Investment = A * C
Y’s Investment = B * D
Profit earned by X = (A*C/ (A*C + B*D)) * P
Profit earned by Y = (B*D/ (A*C + B*D)) * P
So It is nothing but the share earned with respect to the investment made upon the Profit earned by the company. The same applies for 3 workers as well.
Example :
In a Business, A,B,C are three partners who invested Rs 38,40,42 into a business. At the end of year the business earned a profit of Rs 18. What are the profits earned by A,B and C ?
Solution:
Total Profit = 18
Amount Invested by A = 38
Amount Invested by B = 40
Amount Invested by C = 42
Total Investment = 38+40+42= 120
Profit of A = (38/120 ) * 18 = Rs 5.7
Profit of B = 40/120 * 18 = Rs 6
Profit of C = 42/120 *18 = 6.3
Working Partner: The theory of working partner is, he will receive an extra amount/percentage of the total profit earned by the company and then among the rest of the amount (profit amount) that is left, the profit among the other partners and himself will be divided.
Example :
A and B enter into a business by investing 1000 and 2000 respectively. B is a working partner and receives 25% of the profit amount. The business had a profit of 4000. What is the Amount A and B receives?
Solution:
Profit = 4000
B’s Share (since he is the working partner) = 25% of 4000 = 1000
Profit Left = 3000
Profit of A = (1000/3000) * 3000 = 1000
Profit of B = (2000/3000)*3000 = 2000
Total Profit of A = 1000
Total Profit of B = 1000+2000 = 3000
Hence the profit amount shares for A and B are in the ratio 1:3