Unit-1
Ratio, Proportion and percentage
Ratio- The ratio of two quantities ‘x’ and ‘y’ of same kind is obtained by dividing x by y and it is denoted by x : y The first term is called antecedent and the second term is called consequent.
Example: 2/3 or 2:3
Example: Divide 500 into two people A and B ratio 2:3. Sol. Here sum of the ratio is = 2 + 3 = 5 Then, A’s share = B’s share =
Continued ratio- A ratio is said to be in continued if consequent of one ratio is antecedent of the other. For example- a:b and b:c is in continued ratio.
Example: If Rs. 74000 are to be divided among three people X, Y and Z such that Sol. Here X:Y:Z: = 12:15:10 Here sum of the ratio is = 12 + 15 + 10 = 37 Then- X will get = Y will get = Z will get =
Inverse ratio- For the ratio x : y, the inverted ratio will be y:x. Example: inverse ratio of 3:4 is 4:3
Note- A ratio remains same if it is multiplied or divided by the same number. X: y = mx : my ( multiplied by ‘m’) X:y = x/m:y/m (divided by m) Example: 4:5 =
Proportion- Proportion is an equation that equates two ratios in same proportion, If the ratio a/b is equals to the ratio c/d then the following proportion can be written- Here a and d are called the extremes of the proportion, the numbers b and c are called means of the proportion
Properties of proportion-
If then ad = bc 2. Reciprocal property: If two ratios are equal, then their reciprocals are also equal. If then If .
Continued proportion- The quantities p, q, r, s, t..... are said to be in continued proportion of p:q = q:r = r:s.....
Example: If 2, x and 32 are in continued proportion then find the value of x. Sol. Here it can be written as- 2 : x = x:32 Or
Note- if a, b, c are in continued proportion then
Fourth, third and mean proportional- If p:q = r:s then-
Compound proportion- If two or more ratios are multiplied together then they are known as compounded. Thus a1 a2 a3 : b1 b2 b3 is a compounded ratios of the ratios a1 : b1 ; a2 : b2 and a3 : b3. This method is also known as compound rule of three.
Example: 10 men working 8 hours a day can finish a work in 12 days. In how many days can 12 men working 5 hours a day finish the same work?
Sol. This can be arranged as- Then
Important properties of proportion- Given quantities a, b, c, d are in proportion.
Note-
Example: The marks obtained by four examinees are as follows: A : B = 2 : 3, B : C = 4 : 5, C : D = 7 : 9, find the continued ratio. Sol. A : B = 2 : 3 B : C = 4 : 5 = C : D = 7 : 9 = A : B : C : D = 2 : 3 :
Example: Two numbers are in the ratio of 3 : 5 and if 10 be subtracted from each of them, the remainders are in the ratio of 1 : 5, find the numbers. Sol. Let the numbers be x and y, so that- x/y =3/5 or 5x = 3y … (1) Again- Or 5x – y = 40… (2) By solving (1) and(2), we get- X = 12 and y = 20 Hence the required numbers are- 12 and 20
Example: The prime cost of an article was three times the value of material used. The cost of raw materials was increased in the ratio 3 : 4 and the productive wage was increased in the ratio 4 : 5. Find the present prime cost of an article, which could formerly be made for Rs.180.
Sol. Prime cost = x + y, where x = productive wage, y = material used Now prime cost = 180 =3y or, y = 60, again x + y = 180, x = 180–y = 180–60 = 120 Present material cost = 4y/3 and present wage = 5x / 4 Present prime cost– Hence the present prime cost is Rs. 230.
Direct and inverse proportion- Direct proportion or direct variation- Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value. It is represented by the proportional symbol, ∝ Or we can say that two values said to be in direct proportion when one variable increases then other value also increases. If we have two quantities ‘a’ and ‘b’ which are lined with each other directly then we can say- a ∝ b When we remove proportionality sign, the ratio of ‘a’ and ‘b’ becomes equals to a constant. a = Cb For example: if the number of individuals visiting a restaurant increases, earning of the restaurant also increases.
Inverse proportion- Inverse proportion is when one value increases and the other value decreases.
For example- We need more manpower to reduce the time to complete a task. They are inversely proportional.
Questions based on direct and inverse proportion-
Example: A machine in a drinking water supplying factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Sol. Let the bottles filled by machine in 5 hours be x We can write the given information as follows-
There is a direct relationship between time taken and bottles filled by machine, So that, we get- So that 700 bottles can be filled by the machine in 5 hrs.
Example: In a model of a ship, the mast is 9 m high, while the mast of the actual ship is 12 m high. If the length of the ship is 28 m, how long is the model ship? Sol. Suppose the mast of the model ship be x, then We can write the given information as follows-
There is a direct relationship- So that, So that the length of a model ship is 21m. Example: 10 men can do a piece of work in 30 days then in how many days 20 men can do the same work? Sol. Let ‘x’ days be taken to do the task, Then
There is inverse relationship, so that- So that 20 men can do the same piece of work in 15 days.
Key takeaways-
a = Cb 12. Inverse proportion is when one value increases and the other value decreases.
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Percent means a part per 100. A percentage is a number that can be expressed as fraction of 100. We use the symbol % for percentage. We can represent percentage in decimal or fraction form. We use the following formula to calculate percentage- For example: if a student got 300 marks out of 500 then his result in percentage will be-
Percentage Increase and Decrease The percentage increase is equal to the subtraction of original number from a new number, divided by the original number and multiplied by 100. % increase = [(New number – Original number)/Original number] x 100 where, increase in number = New number – original number
% decrease = [(Original number – New number)/Original number] x 100 Where decrease in number = Original number – New number
Example: Find 70 % of 30 Sol.
Example: A person has a monthly salary of Rs. 20,000. He spends 8000 per month on food then what percent of his monthly salary does he save? Sol. Person’s monthly salary = 20,000 Savings = (20,000 – 8000) = 12,000 Fraction of his saving = 12,000/20,000 By converting the fraction into percentage, we get-
Example: A shopkeeper reduced the price of an item by 25%, the old price was 120 then find out the new price. Sol. 25% of 120 will be- Which means he reduced Rs. 30, then the new price is- 120 – 30 = 90 The new price is Rs. 90
Example: If 20% of any number is 500 then find out the number? Sol. Let the number is ‘x’, then- Hence the number is 2500.
decrease in number = Original number – New number
Key takeaways-
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References-
- Mathematics and Statistics for Business – R. S. Bhardwaj – Excel Books.
- Business Mathematics and Statistics – Subhanjali Chopra – Pearson publication.
- Fundamentals of Business Mathematics and Statistics – ICAI – ICAI.
- Business Mathematics and Statistics – Dr. J K Das, N Das – McGraw Hill Education.
- Mathematical and statistical techniques, Dr. Abhilasha S. Magar & Manohar B. Bhagirath