Civil Services Examination - Basic numeracy: ratio proportion profit and loss time and work E-Book Study Material

Basic Numeracy: Ratio, Proportion, Profit and Loss, Time and Work

1. Ratio and Proportion

Ratios and proportions are fundamental concepts in mathematics that help us compare quantities and understand relationships between them. They are widely used in various fields, including finance, engineering, and everyday life.

1.1 Ratio

A ratio is a comparison of two quantities by division. It tells us how many times one quantity contains another. A ratio of 'a' to 'b' can be written as a:b or a/b.

Example: If a class has 20 boys and 30 girls, the ratio of boys to girls is 20:30, which can be simplified to 2:3. This means for every 2 boys, there are 3 girls.

1.2 Types of Ratios

Continued Ratio: A ratio involving three or more quantities, expressed in a way that common terms cancel out. For example, if A:B = 2:3 and B:C = 4:5, the continued ratio A:B:C is found by making the 'B' term common. Multiply the first ratio by 4 and the second by 3 to get A:B = 8:12 and B:C = 12:15. Thus, A:B:C = 8:12:15.
Duplicate Ratio: The ratio of the squares of the terms. The duplicate ratio of a:b is a²:b².
Sub-duplicate Ratio: The ratio of the square roots of the terms. The sub-duplicate ratio of a:b is √a:√b.
Triplicate Ratio: The ratio of the cubes of the terms. The triplicate ratio of a:b is a³:b³.
Sub-triplicate Ratio: The ratio of the cube roots of the terms. The sub-triplicate ratio of a:b is ³√a:³√b.
Invertendo Ratio: The ratio obtained by inverting the terms. The invertendo ratio of a:b is b:a.
Componendo Ratio: If a/b = c/d, then (a+b)/b = (c+d)/d. The ratio is (a+b):b.
Dividendo Ratio: If a/b = c/d, then (a-b)/b = (c-d)/d. The ratio is (a-b):b.
Componendo and Dividendo Ratio: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d). The ratio is (a+b):(a-b).

1.3 Proportion

A proportion is an equation stating that two ratios are equal. If a:b = c:d, then a, b, c, and d are said to be in proportion. This can be written as a/b = c/d.

In the proportion a:b::c:d, 'a' and 'd' are called the extremes, and 'b' and 'c' are called the means. The product of the extremes is equal to the product of the means: a × d = b × c.

1.4 Types of Proportion

Direct Proportion: Two quantities are said to be in direct proportion if they increase or decrease together in the same ratio. If x is directly proportional to y (x ∝ y), then x = ky, where k is a constant.
Inverse Proportion: Two quantities are said to be in inverse proportion if, as one quantity increases, the other decreases in the same ratio, and vice versa. If x is inversely proportional to y (x ∝ 1/y), then xy = k, where k is a constant.

1.5 Problems on Ratio and Proportion

These problems often involve finding unknown values, simplifying ratios, or dividing quantities according to a given ratio. Understanding the relationship between terms and applying the properties of ratios and proportions is key.

Example: The ratio of two numbers is 3:5. If the sum of the numbers is 80, find the numbers.

Let the numbers be 3x and 5x.

3x + 5x = 80

8x = 80

x = 10

The numbers are 3 × 10 = 30 and 5 × 10 = 50.

Shortcut for Continued Ratios: When converting A:B and B:C to A:B:C, find the Least Common Multiple (LCM) of the 'B' terms in both ratios. Adjust the ratios by multiplying each term by the appropriate factor to make the 'B' term equal to the LCM. Then combine them. Example: A:B = 2:3, B:C = 5:7. LCM of 3 and 5 is 15. A:B = (2*5):(3*5) = 10:15 B:C = (5*3):(7*3) = 15:21 So, A:B:C = 10:15:21.

2. Profit and Loss

Profit and loss are essential concepts in business and commerce, dealing with the financial outcome of buying and selling goods or services. They are calculated based on the cost price and selling price.

2.1 Key Terms

Cost Price (CP): The price at which an article is bought.
Selling Price (SP): The price at which an article is sold.
Profit: When SP > CP, the difference is profit. Profit = SP - CP.
Loss: When CP > SP, the difference is loss. Loss = CP - SP.
Overheads: Expenses incurred in bringing an article to a saleable condition (e.g., transportation, repairs, labour). These are added to the CP.

2.2 Calculating Profit and Loss Percentages

Profit and loss percentages are always calculated on the Cost Price (CP), unless stated otherwise.

Profit Percentage: (Profit / CP) × 100
Loss Percentage: (Loss / CP) × 100

2.3 Formulas

If there is a profit of P%, then SP = CP × (100 + P)/100
If there is a loss of L%, then SP = CP × (100 - L)/100
From the above, we can derive formulas for CP:
CP = SP × 100 / (100 + P) (when profit is given)
CP = SP × 100 / (100 - L) (when loss is given)

2.4 Problems involving Discount

Discount is a reduction in the price of an article, usually offered by the seller to the buyer. It is calculated on the Marked Price (MP).

Marked Price (MP): The price listed on the article, often higher than the CP.
Discount: The reduction allowed on the MP. Discount = MP - SP.
Discount Percentage: (Discount / MP) × 100

Relationship: MP - Discount = SP

MP × (100 - Discount %)/100 = SP

2.5 Example Calculation

An item is bought for ₹400 and sold for ₹480.

CP = ₹400

SP = ₹480

Since SP > CP, there is a profit.

Profit = SP - CP = ₹480 - ₹400 = ₹80

Profit Percentage = (Profit / CP) × 100 = (80 / 400) × 100 = (1/5) × 100 = 20%

Now, suppose the item was marked at ₹600 and sold for ₹480.

MP = ₹600

SP = ₹480

Discount = MP - SP = ₹600 - ₹480 = ₹120

Discount Percentage = (Discount / MP) × 100 = (120 / 600) × 100 = (1/5) × 100 = 20%

Profit/Loss Shortcut: If you buy two items for the same total cost price, and sell one at a profit of x% and the other at a loss of x%, then there is always an overall loss of (x/10)²%. Example: Two tables are bought for ₹1000 each. One is sold at a profit of 20%, and the other at a loss of 20%. Total CP = ₹2000. SP1 = 1000 * (120/100) = ₹1200. SP2 = 1000 * (80/100) = ₹800. Total SP = ₹2000. Net profit/loss = 0. *Correction*: This shortcut applies when the SELLING PRICES are the same. If the cost prices are the same, and selling prices differ by x%, it's different. Let's use a common scenario: Two items are sold for the SAME SELLING PRICE. One is sold at a profit of x%, the other at a loss of x%. There is always a NET LOSS of (x/10)²%. Example: Two articles are sold for ₹1000 each. One at a profit of 20%, the other at a loss of 20%. SP1 = ₹1000 (Profit 20%) => CP1 = 1000 * (100/120) = ₹833.33 SP2 = ₹1000 (Loss 20%) => CP2 = 1000 * (100/80) = ₹1250 Total SP = 1000 + 1000 = ₹2000 Total CP = 833.33 + 1250 = ₹2083.33 Net Loss = 2083.33 - 2000 = ₹83.33 Net Loss % = (83.33 / 2083.33) * 100 ≈ 4%. Using the shortcut: Loss % = (20/10)² = 2² = 4%. This shortcut is very useful!

3. Time and Work

The Time and Work section deals with problems involving the rate at which individuals or groups complete tasks. It's based on the principle that more workers or more time generally leads to more work done.

3.1 Basic Concepts

Work: The total task to be completed. It is often considered as 1 unit of work.
Rate of Work: The amount of work done by a person or group in one unit of time (e.g., per day, per hour).
If a person can complete a work in 'D' days, their rate of work is 1/D per day.
If a person's rate of work is 'R' per day, the total days required to complete the work is 1/R.

3.2 Relationship between Work, Rate, and Time

Work = Rate × Time

This fundamental formula underpins all Time and Work problems.

3.3 Scenarios and Formulas

Single Person: If A can do a piece of work in 'a' days, then A's rate is 1/a work per day.
Multiple People Working Together: If A can do a work in 'a' days and B can do the same work in 'b' days, then:

A's rate = 1/a
B's rate = 1/b
Combined rate (A+B) = A's rate + B's rate = 1/a + 1/b = (a+b)/ab
Time taken by (A+B) together = 1 / (Combined rate) = ab / (a+b) days.

Men, Days, and Work: The principle M₁D₁/W₁ = M₂D₂/W₂ is used, where M is the number of men, D is the number of days, and W is the amount of work. This assumes the rate of work per man is constant.
Efficiency: If A is 'x' times more efficient than B, it means A can do 'x' times the work B can do in the same time. If B takes 'd' days, A will take d/x days. The ratio of their efficiencies is the inverse ratio of the time taken. Efficiency ratio A:B = Time ratio B:A.
Work and Wages: The ratio of wages earned by different people working together is directly proportional to the ratio of their individual rates of work.

3.4 Example Calculation

A can complete a piece of work in 10 days. B can complete the same work in 15 days. In how many days can they complete the work together?

A's rate = 1/10 work per day.

B's rate = 1/15 work per day.

Combined rate = 1/10 + 1/15

To add these fractions, find a common denominator, which is 30.

Combined rate = 3/30 + 2/30 = 5/30 = 1/6 work per day.

Time taken together = 1 / (Combined rate) = 1 / (1/6) = 6 days.

Alternative using MxD formula:

Let A do the work in D_A = 10 days and B do it in D_B = 15 days.

Time together = (D_A × D_B) / (D_A + D_B)

Time together = (10 × 15) / (10 + 15) = 150 / 25 = 6 days.

Time and Work Shortcut: For problems involving multiple people working together, think in terms of LCM. If A takes 10 days and B takes 15 days, the LCM of 10 and 15 is 30. Assume the total work is 30 units. A's efficiency = 30 units / 10 days = 3 units/day. B's efficiency = 30 units / 15 days = 2 units/day. Combined efficiency = 3 + 2 = 5 units/day. Time taken together = Total Work / Combined Efficiency = 30 units / 5 units/day = 6 days. This LCM method simplifies calculations, especially when days are not easily divisible.

3.5 Work Done by Different Efficiencies

If A and B can complete a work in 'x' days and 'y' days respectively, and C is 'k' times as efficient as A and B combined:

Combined rate of A and B = 1/x + 1/y = (x+y)/xy

Rate of C = k × (1/x + 1/y)

Time taken by C alone = 1 / (Rate of C)

3.6 Men, Machines, and Time

Problems can involve men, women, children, or machines working at different rates. The principle M₁D₁H₁/W₁ = M₂D₂H₂/W₂ is applicable, where H represents hours per day.

Example: 10 men can complete a work in 15 days. In how many days will 6 men complete the same work?

Using M₁D₁ = M₂D₂:

10 men × 15 days = 6 men × D₂ days

150 = 6 × D₂

D₂ = 150 / 6 = 25 days.

3.7 Negative Work

This involves tasks where some entities add to the work (like filling a tank) and others remove from it (like emptying a tank). The rates are added for filling and subtracted for emptying.

Example: Pipe A can fill a tank in 10 hours. Pipe B can empty it in 15 hours. If both pipes are opened simultaneously, how long will it take to fill the tank?

Rate of A (filling) = +1/10 tank per hour.

Rate of B (emptying) = -1/15 tank per hour.

Net rate = 1/10 - 1/15 = 3/30 - 2/30 = 1/30 tank per hour.

Time to fill the tank = 1 / (Net rate) = 1 / (1/30) = 30 hours.

Chronological Thinking for Time & Work: Always think of the total work as a 'whole pie'. People take 'slices' out of the pie per day. If someone is emptying, they are putting slices back into the box, reducing the amount eaten. Calculate the net 'slices eaten per day' to find the total time.