SSC CGL Tier 1 2011 Quantitative Aptitude Questions with Solutions

Explanation

The area of the triangular field is calculated using Heron's formula. Given the sides a = 26m, b = 28m, and c = 30m, the semi-perimeter s is (26 + 28 + 30) / 2 = 42m. The area A is given by √(s(s-a)(s-b)(s-c)) = √(42(42-26)(42-28)(42-30)) = √(42 × 16 × 14 × 12). Simplifying this, A = √((14 × 3) × 16 × 14 × (4 × 3)) = 14 × 3 × 4 × 2 = 336 m².

The three cows are tethered at the corners, and the area each cow grazes forms a sector of a circle with radius r = 7m. Since the sum of the internal angles of any triangle is 180°, the combined area of the three sectors is equal to the area of a semicircle. Total grazed area is ½ π r² = ½ × ²²⁄₇ × 7² = 77 m².

The ungrazed area is the total area of the triangle minus the total grazed area, which is 336 - 77 = 259 m².

Explanation

The gain of 56 represents a 10% profit on the cost price (CP). Using the relationship Gain = Profit % × CP, we have 56 = 0.10 × CP, which gives CP = 560. The selling price (SP) is the sum of the cost price and the gain, calculated as SP = 560 + 56 = 616. The shopkeeper allows a 23% commission on the advertised price (MP), meaning the selling price is 100% - 23% = 77% of the advertised price. This is expressed by the equation 0.77 × MP = 616. Solving for the advertised price, MP = 616 / 0.77 = 800.

Explanation

To determine the single equivalent discount for three successive discounts of 10%, 12%, and 5%, we calculate the final price relative to an initial price of 100. After the first discount of 10%, the price becomes 100 × (1 - 0.10) = 90. Applying the second discount of 12% to this result gives 90 × (1 - 0.12) = 90 × 0.88 = 79.2. Finally, applying the third discount of 5% yields 79.2 × (1 - 0.05) = 79.2 × 0.95 = 75.24. The single equivalent discount is the total reduction from the original price, which is 100 - 75.24 = 24.76%.

This can also be solved using the successive discount formula x + y - ˣʸ⁄₁₀₀. Applying it to the first two discounts gives 10 + 12 - (10 × 12)/(100) = 20.8%. Combining this result with the third discount gives 20.8 + 5 - (20.8 × 5)/(100) = 25.8 - 1.04 = 24.76%.

Explanation

Given the ratio x:y = 5:6, we can assign x = 5k and y = 6k for some constant k. The expression (3x - 2y) : (y - x) can be written in fraction form and evaluated by substituting these values:

(3x - 2y)/(y - x) = (3(5k) - 2(6k))/(6k - 5k)

(15k - 12k)/(k) = ³ᵏ⁄ₖ

³⁄₁

This simplifies to the ratio 3:1.

Explanation

The initial ratio of copper, zinc, and nickel is 5:3:2. The sum of these ratio parts is 5 + 3 + 2 = 10. Given that the total weight of the alloy is 100 kg, the weight corresponding to one ratio part is 100 / 10 = 10 kg.

The initial quantity of nickel in the alloy is 2 × 10 = 20 kg.

The new ratio after adding nickel is 5:3:3. Since only nickel is added, the ratio parts for copper (5) and zinc (3) remain unchanged relative to each other and their absolute weights stay the same. In the new ratio, the nickel component increases from 2 parts to 3 parts, an increase of 1 part. Since 1 ratio part is equal to 10 kg, the quantity of nickel that must be added is 10 kg. Alternatively, the new total quantity of nickel is 3 × 10 = 30 kg, and the amount added is 30 - 20 = 10 kg.

Explanation

Let the ages of Ram and Rahim 10 years ago be x and 3x respectively. Their present ages are (x + 10) and (3x + 10).

Five years from the present, their ages will be (x + 15) and (3x + 15). According to the given condition:

(x + 15)/(3x + 15) = ⅔

By cross-multiplying the terms: 3(x + 15) = 2(3x + 15) 3x + 45 = 6x + 30 15 = 3x x = 5

Substituting the value of x into the expressions for their present ages: Ram's present age = 5 + 10 = 15 years Rahim's present age = 3(5) + 10 = 25 years

The ratio of their present ages is 15:25. Dividing both terms by their greatest common divisor, 5, we get the simplified ratio of 3:5.

Explanation

Let the incomes of A, B, and C be 7x, 9x, and 12x respectively. Let their expenditures be 8y, 9y, and 15y respectively. According to the problem, A saves 1/4th of his income, which means his expenditure is 3/4ths of his income.

We can set up the equation for A as 8y = ¾ × 7x. Simplifying this, we get 32y = 21x, which means x = ³²ʸ⁄₂₁. Substituting this value of x to find the ratio of their incomes in terms of y: A's income = 7 × ³²ʸ⁄₂₁ = ³²ʸ⁄₃ B's income = 9 × ³²ʸ⁄₂₁ = ⁹⁶ʸ⁄₇ C's income = 12 × ³²ʸ⁄₂₁ = ¹²⁸ʸ⁄₇

Savings are calculated as Income minus Expenditure: A's savings = ³²ʸ⁄₃ - 8y = (32y - 24y)/(3) = ⁸ʸ⁄₃ B's savings = ⁹⁶ʸ⁄₇ - 9y = (96y - 63y)/(7) = ³³ʸ⁄₇ C's savings = ¹²⁸ʸ⁄₇ - 15y = (128y - 105y)/(7) = ²³ʸ⁄₇

To find the ratio A:B:C, we divide their savings: ⁸⁄₃ : ³³⁄₇ : ²³⁄₇ Multiplying the entire ratio by the least common multiple of the denominators, which is 21: 8 × 7 : 33 × 3 : 23 × 3 56 : 99 : 69

Explanation

The total sum of the four numbers is calculated by multiplying the average by the count: 60 × 4 = 240. Let the first number be x and the sum of the remaining three numbers be S. According to the problem, the first number is one-fourth of the sum of the last three, which is expressed as x = ¼S, or S = 4x.

The total sum of all four numbers can be written as the sum of the first number and the sum of the other three: x + S = 240. Substituting S = 4x into this equation gives x + 4x = 240, which simplifies to 5x = 240. Solving for x, we find x = ²⁴⁰⁄₅ = 48.

Explanation

The new average is calculated by accounting for the difference between the correct observation and the incorrect one.

Difference = Correct Value - Incorrect Value = 73 - 48 = 25

Since the sum of observations increases by 25, the increase in the average is distributed across all 25 observations.

Increase in Average = (Total Difference)/(Number of Observations) = ²⁵⁄₂₅ = 1

New Average = Old Average + Increase in Average = 13 + 1 = 14

Explanation

Let the cost price (CP) of one article be x. According to the problem, the cost price of 10 articles is equal to the selling price (SP) of 8 articles. Therefore, the SP of 8 articles is 10x. To find the selling price of a single article, we divide the total selling price by the number of articles: SP of 1 article = 10x / 8 = 1.25x.

The gain made on each article is the difference between its selling price and its cost price: Gain = 1.25x - x = 0.25x. To express this as a percentage, we use the formula Gain % = (Gain / CP) × 100. Substituting the values into the formula yields Gain % = (0.25x / x) × 100 = 25%.

Alternatively, for problems where the CP of m articles equals the SP of n articles, the gain or loss percentage can be calculated using the formula Gain % = [(m - n) / n] × 100. Substituting m = 10 and n = 8, we get [(10 - 8) / 8] × 100 = (2 / 8) × 100 = (1 / 4) × 100 = 25%.

Explanation

The net gain or loss percentage when there is a markup followed by a discount can be calculated using the successive percentage change formula: x + y + ˣʸ⁄₁₀₀.

In this case, the markup is x = 40% and the discount is y = -30%. Substituting these values into the formula gives:

Net Change = 40 - 30 + (40 × (-30))/(100)

Net Change = 10 - ¹²⁰⁰⁄₁₀₀

Net Change = 10 - 12 = -2%

The negative sign indicates a loss of 2%. Alternatively, if we assume the Cost Price is 100, the Marked Price becomes 140 after a 40% markup. A 30% discount on 140 is 42, which makes the Selling Price 140 - 42 = 98. Since the Selling Price is 2 less than the Cost Price, there is a 2% loss.

Explanation

To find the number of oranges required to earn a specific profit, we first determine the cost price and selling price per orange. The cost price of 8 oranges is 34, which means the cost price of a single orange is 34 / 8 = 4.25. The selling price of 12 oranges is 57, making the selling price of a single orange 57 / 12 = 4.75. The profit earned on each individual orange is the difference between the selling price and the cost price, calculated as 4.75 - 4.25 = 0.50. To achieve a total net profit of 45, we divide the desired total profit by the profit earned per orange. This gives 45 / 0.50 = 90.

Alternatively, using the least common multiple of the quantities 8 and 12, which is 24, we can compare the transactions for 24 oranges. The cost price for 24 oranges is 3 × 34 = 102, and the selling price for 24 oranges is 2 × 57 = 114. This results in a profit of 114 - 102 = 12 for every 24 oranges sold. To find the quantity needed for a profit of 45, we use the ratio 24 × (45 / 12) = 90.

Explanation

The total selling price of the two articles is 5000 + 5000 = 10000. Since the man realizes no gain and no loss on the entire transaction, the total cost price of both articles must also be 10000.

For the first article, the selling price is 5000 and the gain is 25%. The cost price of this article is calculated using the formula CP₁ = (SP × 100)/(100 + Gain%) = (5000 × 100)/(125) = 4000. The profit made on the first article is 5000 - 4000 = 1000.

To ensure there is no overall profit or loss, the profit of 1000 from the first article must be offset by a loss of 1000 on the second article. For the second article, the selling price is 5000 and the loss is 1000, so the cost price is CP₂ = SP + Loss = 5000 + 1000 = 6000.

The loss percentage on the second article is calculated as (Loss)/(CP₂₎ × 100 = ¹⁰⁰⁰⁄₆₀₀₀ × 100 = ¹⁰⁰⁄₆ = 16 ⅔%.

Explanation

Let the cost price of the article be x. According to the problem, the percentage gain is also x%. The relationship between selling price (SP), cost price (CP), and gain percentage is given by the formula SP = CP × (1 + (Gain%)/(100)).

Substituting the given values into the formula: 144 = x × (1 + ˣ⁄₁₀₀) 144 = (100x + x²)/(100) x² + 100x - 14400 = 0

To solve this quadratic equation, we factorize the middle term: x² + 180x - 80x - 14400 = 0 x(x + 180) - 80(x + 180) = 0 (x - 80)(x + 180) = 0

This gives x = 80 or x = -180. Since the cost price cannot be negative, we take x = 80. Alternatively, checking the options, if CP = 80, the gain is 80% of 80, which is 64. The SP = 80 + 64 = 144, which matches the given information.

Explanation

The total percentage given away by Meera is 15% + 32% + 25% = 72%. The remaining percentage of her total amount is 100% - 72% = 28%.

Let the total amount Meera had be x. According to the problem, ¾ of the remaining amount is 8,400. We can set up the equation: ¾ × ²⁸⁄₁₀₀ × x = 8,400

Simplifying the equation: (3 × 7)/(100) × x = 8,400 ²¹⁄₁₀₀ × x = 8,400 x = (8,400 × 100)/(21) x = 40,000

The total amount Meera had was 40,000. The amount she gave to Anish was 15% of this total: Anish's share = ¹⁵⁄₁₀₀ × 40,000 = 6,000

Explanation

The amount saved due to the 20% price reduction on the total expenditure of 54 is calculated as 0.20 × 54 = 10.8 . This saved amount of 10.8 is exactly what pays for the 10 additional apples. Thus, the reduced price of one apple is (10.8)/(10) = 1.08 . To find the reduced price per dozen, we multiply the price of a single apple by 12. Therefore, the reduced price per dozen is 1.08 × 12 = 12.96 .

Explanation

The expenditure on a commodity is the product of its price and the quantity consumed. To keep the expenditure constant when the price increases by r%, the consumption must be reduced by the formula (r)/(100 + r) × 100%. Here, the price increase r is 60%. Substituting this value into the formula, the reduction percentage is (60)/(100 + 60) × 100, which simplifies to ⁶⁰⁄₁₆₀ × 100. Reducing the fraction ⁶⁰⁄₁₆₀ gives ⅜. Multiplying ⅜ by 100 results in 37.5%. This can also be understood by assuming the initial price is 10 and consumption is 10, making the expenditure 100. An increase of 60% makes the new price 16. For the expenditure to remain 100, the new consumption must be ¹⁰⁰⁄₁₆, which equals 6.25. The reduction from the original consumption of 10 is 10 - 6.25 = 3.75, which is 37.5% of the original amount.

Explanation

This problem can be solved using the principle of weighted averages or simple distributive properties. Let A, B, and G represent the initial cost of an apple, a banana, and a guava, respectively. The initial total cost of the fruits is T = 4B + 2A + 3G.

According to the problem, the cost of each type of fruit increases by 10%. Therefore, the new prices are 1.10A, 1.10B, and 1.10G. The new total cost T' is calculated as follows:

T' = 4(1.10B) + 2(1.10A) + 3(1.10G)

By factoring out the common multiplier of 1.10 from each term, we get:

T' = 1.10(4B + 2A + 3G)

T' = 1.10T

The percentage increase in the total cost is given by the formula:

Percentage Increase = (T' - T)/(T) × 100

Substituting T' = 1.10T:

Percentage Increase = (1.10T - T)/(T) × 100 = (0.10T)/(T) × 100 = 10%

Whenever every individual component in a linear combination increases by the same percentage, the total sum of that combination will also increase by that same percentage, regardless of the individual quantities or original prices of the components.

Explanation

For a constant distance, speed and time are inversely proportional (S ∝ 1/T). When the speed becomes 6/7 of the usual speed, the time taken becomes the reciprocal, which is 7/6 of the usual time.

Let the usual time be T. According to the problem, the difference between the new time and the usual time is 12 minutes: ⁷⁄₆T - T = 12 ⅙T = 12 T = 72 minutes

To convert 72 minutes into hours and minutes, we divide by 60: 72 minutes = 1 hour and 12 minutes

Explanation

The distance d between two points when a person travels at different speeds v₁ and v₂ and arrives with a specific time difference Δ t can be calculated using the formula d = (v₁ · v₂)/(v₂ - v₁) · Δ t. In this scenario, the two speeds are v₁ = 5 km/hour and v₂ = 6 km/hour. The total time difference Δ t is the interval between being 7 minutes late and 5 minutes early, which is 7 + 5 = 12 minutes. Since speed is in km/hour, the time difference must be converted to hours: Δ t = ¹²⁄₆₀ = ⅕ hour. Substituting these values into the formula yields d = (5 · 6)/(6 - 5) · ⅕. Simplifying the expression results in d = ³⁰⁄₁ · ⅕ = 6 km.

Explanation

The initial rate of interest R is calculated using the simple interest formula SI = (P × R × T)/(100). Given that 800 becomes 956 in 3 years, the interest earned is 956 - 800 = 156. Substituting the values into the formula:

156 = (800 × R × 3)/(100) 156 = 24R R = ¹⁵⁶⁄₂₄ = 6.5%

The new rate of interest is increased by 4%, so the updated rate is 6.5% + 4% = 10.5%. To find the amount 1800 will become in 3 years at this new rate, we first calculate the new interest:

SIₙₑ_w = (1800 × 10.5 × 3)/(100) SIₙₑ_w = 18 × 31.5 = 567

The final amount is the sum of the principal and the interest: Amount = 1800 + 567 = 2367

Explanation

The standard formula for simple interest is SI = (P × R × T)/(100), where P is the principal sum, R is the rate of interest per annum, and T is the time in years. The problem states that SI = ¹⁶⁄₂₅P and that the numerical values of the rate and time are equal, so R = T.

Substituting these values into the formula gives ¹⁶⁄₂₅P = (P × R × R)/(100). By canceling P from both sides, the equation becomes ¹⁶⁄₂₅ = R²{100}. Rearranging to solve for R², we get R² = (16 × 100)/(25) = 16 × 4 = 64. Taking the square root of both sides yields R = √(64) = 8. Thus, the rate percent is 8%.

Explanation

The difference between Compound Interest (C.I.) and Simple Interest (S.I.) for a period of 2 years is calculated using the formula D = P ( ʳ⁄₁₀₀ )², where P is the principal sum and r is the annual rate of interest. Given that the difference D is 6 and the rate r is 5%, the equation becomes 6 = P ( ⁵⁄₁₀₀ )².

Simplifying the fraction inside the parentheses gives 6 = P ( ¹⁄₂₀ )². Squaring the term results in 6 = P × ¹⁄₄₀₀. By cross-multiplying to solve for the principal, P = 6 × 400, which equals 2400.