SSC CHSL Tier 1 2010 Quantitative Aptitude Questions with Solutions

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

In ΔABC, AD is the bisector of angle A meeting BC at D. Therefore, the relation is:

BD/DC = AB/AC

Substituting the given values, AB = 3 cm and AC = 4 cm:

BD/DC = 3/4

This gives the ratio BD : DC as 3 : 4.

Explanation

Let the side of the square be a. The area of the square is given as: a² = 121 cm² a = √121 = 11 cm

The perimeter of the square, which is the total length of the copper wire, is: Perimeter = 4a = 4 × 11 = 44 cm

When the same wire is bent into a circle of radius r, the circumference of the circle equals the length of the wire: 2πr = 44 2 × (22/7) × r = 44 r = 7 cm

The area of the circle is: Area = πr² = (22/7) × 7² = (22/7) × 49 = 154 cm²

Explanation

Let the length, width, and height of the cuboid be l, w, and h respectively.

The volume V of the cuboid is given by the formula: V = lwh

The surface areas of the three adjacent faces of the cuboid are: p = lw q = wh r = hl

Multiplying these three equations together: p × q × r = (lw) × (wh) × (hl) pqr = l²w²h² pqr = (lwh)²

Since V = lwh, we can substitute V into the equation: pqr = V²

Taking the square root on both sides gives: V = √(pqr)

Explanation

The given sequence follows an alternating pattern of adding 4 and subtracting 3 to obtain successive terms:

8 + 4 = 12 12 - 3 = 9 9 + 4 = 13 13 - 3 = 10 10 + 4 = 14

Following this consistent rule, the next operation requires subtracting 3 from the sixth term:

14 - 3 = 11

Alternatively, the sequence can be split into two alternating arithmetic progressions: The terms at odd positions are 8, 9, 10, ... which increase by 1 each time. The terms at even positions are 12, 13, 14, ... which also increase by 1 each time.

The next term in the sequence is at the seventh position, which is an odd position. It follows the first progression, giving 10 + 1 = 11.

Explanation

The given sequence consists of consecutive prime numbers starting from 3. The prime numbers in this range are 3, 5, 7, 11, 13, 17, and 19. The number 9 is a composite number because it is divisible by 3, which makes it the wrong number in this prime number sequence.

Explanation

The rate of the filling tap is 1/40 of the tank per minute, while the rate of the emptying tap is 1/60 of the tank per minute.

When both taps are opened simultaneously, the net filling rate per minute is the difference between the filling rate and the emptying rate: Net rate = 1/40 - 1/60 = (3 - 2) / 120 = 1/120 of the tank per minute

This means the entire tank will be filled in 120 minutes.

To convert this duration into hours: Time = 120 / 60 = 2 hours

Explanation

Let the total work be represented as 1 unit.

The combined rate of work of A, B, and C is 1/30 of the work per minute. The combined rate of work of A and B is 1/50 of the work per minute.

The rate of work of C alone is the difference between these two rates: Rate of C = 1/30 - 1/50

Taking the lowest common multiple of 30 and 50, which is 150, the calculation is: Rate of C = 5/150 - 3/150 = 2/150 = 1/75 of the work per minute

The time taken by C alone to complete the work is the reciprocal of this rate: Time = 1 / (1/75) = 75 minutes

Explanation

To find who completes the work first, we calculate the total time each person takes to complete 100% of the work.

For P, 1/4 of the work is completed in 10 days, so the entire work is completed in 10 × 4 = 40 days.

For Q, 40% (or 2/5) of the work is completed in 15 days, so the entire work is completed in 15 × 5/2 = 37.5 days.

For R, 1/3 of the work is completed in 13 days, so the entire work is completed in 13 × 3 = 39 days.

For S, 1/6 of the work is completed in 7 days, so the entire work is completed in 7 × 6 = 42 days.

Comparing the total days, Q requires the least number of days (37.5 days) to complete the work.

Explanation

Let the side of the square be s and the radius of the circle be r.

The area of a square is s² and the area of a circle is πr². Since the area of the square and the circle are equal, we write the relation: s² = πr²

Taking the square root on both sides gives the relation between the side and the radius: s = √π r

Dividing both sides by r yields the ratio of the side of the square to the radius of the circle: s/r = √π / 1

This gives the ratio of √π : 1.

Explanation

Let the monthly salaries of A, B, and C be 2x, 3x, and 5x respectively.

The difference between the monthly salaries of C and A is: 5x - 2x = 3x

According to the given information, this difference is Rs. 12,000: 3x = 12,000

This means B's monthly salary, which is represented by 3x, is exactly Rs. 12,000.

To find B's annual salary, multiply B's monthly salary by 12: 12,000 × 12 = 1,44,000

Thus, B's annual salary is Rs. 1,44,000.

Explanation

To find the combined ratio A:B:C:D:E, we first simplify the given ratios where possible: C:D = 6:9 = 2:3 D:E = 12:16 = 3:4

Since the value of D is 3 in both C:D (2:3) and D:E (3:4), we can directly combine them to obtain: C:D:E = 2:3:4

Next, we combine B:C = 3:4 with C:D:E = 2:3:4. The term C is 4 in B:C and 2 in C:D:E. To make the C terms equal, we multiply the ratio C:D:E by 2, which gives: C:D:E = 4:6:8

Now we can combine B:C with this updated C:D:E to get: B:C:D:E = 3:4:6:8

Finally, we combine A:B = 1:2 with B:C:D:E = 3:4:6:8. The term B is 2 in the first ratio and 3 in the second. The least common multiple of 2 and 3 is 6. We multiply A:B by 3 and B:C:D:E by 2: A:B = 3:6 B:C:D:E = 6:8:12:16

Merging these two ratios yields the final combined ratio: A:B:C:D:E = 3:6:8:12:16

Explanation

Given the ratio x:y = 2:5, let x = 2k and y = 5k.

Substituting these values into the expression:

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

Simplifying the terms:

Numerator = 10k + 15k = 25k Denominator = 10k - 15k = -5k

Dividing the numerator by the denominator:

25k / -5k = -5

Explanation

The slant height l of a right circular cone is given by the relation l = √(r² + h²), where r is the radius of the base and h is the height.

The radius of the base is half of the diameter: r = 4/2 = 2 cm

Substituting the values of radius r = 2 cm and height h = 2√3 cm into the formula: l = √(2² + (2√3)²) l = √(4 + 12) l = √16 l = 4 cm

Explanation

Let the cost price of the goods be ₹100.

Since the dealer marks his goods 20% above the cost price, the marked price is: Marked Price = 100 + 20% of 100 = ₹120

To make a profit of 10%, the selling price must be: Selling Price = 100 + 10% of 100 = ₹110

The discount offered on the marked price is: Discount = Marked Price - Selling Price = 120 - 110 = ₹10

The rate of discount is: Discount Rate = (Discount / Marked Price) × 100% Discount Rate = (10 / 120) × 100% = 100/12% = 25/3% = 8⅓%

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Gain percentage (G), and Discount percentage (D) is given by the formula:

MP/CP = (100 + G) / (100 - D)

In the first scenario, with a 10% discount and a 20% gain:

MP/CP = (100 + 20) / (100 - 10) = 120/90 = 4/3

In the second scenario, with a 20% discount and an unknown gain G:

MP/CP = (100 + G) / (100 - 20)

Since MP/CP remains constant:

4/3 = (100 + G) / 80

320/3 = 100 + G

106 2/3 = 100 + G

G = 6 2/3%

Explanation

The single equivalent discount d for two successive discounts of a% and b% is given by the formula:

d = a + b - (a × b)/100

Substituting the given values, a = 20 and b = 5:

d = 20 + 5 - (20 × 5)/100 d = 25 - 100/100 d = 25 - 1 d = 24%

Explanation

Since only water is added, the quantity of milk remains constant.

Initial volume of milk = 90% of 40 = 36 litres

In the new mixture, water is 20%, which means milk is 80%.

Let the new mixture volume be V. 0.8 × V = 36 V = 36 ÷ 0.8 = 45 litres

Water to be added = 45 - 40 = 5 litres

Explanation

At the time of marriage 4 years ago, the average age of the husband and wife was 25 years. Therefore, the sum of their ages at that time was 25 × 2 = 50 years.

Since both the husband and wife have aged by 4 years, the sum of their present ages today is 50 + 4 + 4 = 58 years.

Today, the family consists of 3 members: the husband, the wife, and the child. The average age of this family of 3 is 20 years. Therefore, the sum of their present ages today is 20 × 3 = 60 years.

The present age of the child is obtained by subtracting the sum of the parents' present ages from the total family age sum today: 60 - 58 = 2 years.

Explanation

Let T₄ be the temperature of the fourth day. The sum of the temperatures of the first four days and the last four days includes the temperature of the fourth day twice. Therefore, the relation is:

T₄ = (Sum of the first 4 days) + (Sum of the last 4 days) - (Sum of all 7 days)

Using the formula for sum, Sum = Average × Number of observations:

Sum of the first 4 days = 4 × 37 = 148 °C Sum of the last 4 days = 4 × 41 = 164 °C Sum of all 7 days of the week = 7 × 39 = 273 °C

Substituting these values into the relation:

T₄ = (148 + 164) - 273 = 312 - 273 = 39 °C

Explanation

The sum of the initial five numbers is calculated by multiplying their average by the count of numbers: Sum of 5 numbers = 5 × 7 = 35

Similarly, the sum of the eight numbers after adding the three new numbers is: Sum of 8 numbers = 8 × 8.5 = 68

The sum of the three new numbers is the difference between the sum of the eight numbers and the sum of the initial five numbers: Sum of 3 new numbers = 68 - 35 = 33

The average of these three new numbers is their sum divided by three: Average of 3 new numbers = 33 / 3 = 11

Explanation

Let the original two-digit number be 10a + b, where a and b are the tens and units digits. When the digits are interchanged, the number becomes 10b + a.

The difference between the original number and the incorrect number is: (10a + b) - (10b + a) = 9(a - b)

Since the average of the ten numbers decreases by 1.8, the total sum of the ten numbers decreases by: 10 × 1.8 = 18

Equating the decrease in the sum to the difference between the numbers: 9(a - b) = 18 a - b = 2

The difference between the digits of the number in which the mistake was made is 2.

Explanation

Let the Cost Price of the article be CP.

When the article is sold for ₹700, there is a loss of 30%. Selling Price = CP × (1 - Loss/100) 700 = CP × (1 - 30/100) 700 = CP × 0.70 CP = 700 / 0.70 = 1000

To obtain a gain of 30%, the required selling price is: New Selling Price = CP × (1 + Gain/100) New Selling Price = 1000 × (1 + 30/100) New Selling Price = 1000 × 1.30 = 1300

Explanation

Let the original price of wheat be P per kg.

The original quantity of wheat purchased for Rs. 320 is 320/P kg.

With a 20% reduction, the new price becomes 80% of P, which is 0.8P per kg.

The new quantity of wheat purchased for the same amount is 320/(0.8P) kg.

According to the question, the difference in quantity is 5 kg:

320/(0.8P) - 320/P = 5

400/P - 320/P = 5

80/P = 5

P = 80/5 = 16

Explanation

Let CP be the cost price of the bicycle.

The difference between the two profit percentages corresponds directly to the increase in the selling price.

Difference in gain percentage = 14% - 8% = 6%

Since this 6% increase in profit margin represents the Rs. 75 increase in the selling price:

6% of CP = 75

(6 / 100) × CP = 75

CP = (75 × 100) / 6

CP = 1250

Explanation

Let the selling price of the bedsheet be SP. The cost price is CP = Rs. 450.

Since the gain is 10% calculated on the selling price: Gain = 10% of SP = 0.10 × SP

Using the relation: SP = CP + Gain SP = 450 + 0.10 × SP 0.90 × SP = 450 SP = 450 / 0.90 SP = 500

Explanation

Total marks of both subjects = 300 + 200 = 500

Overall target marks required = 46% of 500 = (46 / 100) × 500 = 230

Marks scored in Science = 32% of 300 = (32 / 100) × 300 = 96

Marks needed in language papers = 230 - 96 = 134

Percentage marks needed in language papers = (134 / 200) × 100 = 67%

Explanation

When two objects move towards each other on parallel paths, their relative speed is the sum of their individual speeds.

Relative speed = 42 km/hr + 48 km/hr = 90 km/hr

To convert this relative speed from kilometres per hour to metres per second, we multiply by the factor 5/18: Relative speed = 90 × 5/18 = 25 m/s

The total distance to be covered for the two trains to completely cross each other is the sum of their respective lengths: Total distance = 137 m + 163 m = 300 m

Using the fundamental speed-distance-time formula, the time taken to cross each other is: Time = Total distance ÷ Relative speed = 300 ÷ 25 = 12 seconds

Explanation

The reduced price per kg is directly calculated using the formula:

Reduced Price = (Total Expenditure × Percentage Reduction) / (Additional Quantity × 100)

Substituting the given values:

Reduced Price = (600 × 25) / (10 × 100) = 15000 / 1000 = 15

The reduction in price saves 25% of Rs. 600, which is Rs. 150. This saved amount of Rs. 150 is what allows the person to buy 10 kg more rice, which means the reduced price of rice per kg is Rs. 150 divided by 10 kg, equaling Rs. 15 per kg.

Explanation

When the price of an article is reduced by r%, the percentage increase required to restore the original price is given by the formula:

Percentage Increase = (r / (100 - r)) × 100%

Given that the price is reduced by 25%, substituting r = 25 into the formula gives:

Percentage Increase = (25 / (100 - 25)) × 100% Percentage Increase = (25 / 75) × 100% Percentage Increase = ⅓ × 100% Percentage Increase = 33⅓%

This represents the exact percentage by which the new price must be increased to return to the original value.

Explanation

To restore a value that has been decreased by x% back to its original level, the required percentage increase is given by the formula:

Percentage Increase = (x / (100 - x)) × 100

Here, the production has decreased by 20%, so x = 20. Substituting this value into the formula:

Percentage Increase = (20 / (100 - 20)) × 100 Percentage Increase = (20 / 80) × 100 Percentage Increase = ¼ × 100 Percentage Increase = 25%

Explanation

The formula for simple interest is SI = (P × R × T) / 100, where P is the principal sum, R is the annual rate of interest, and T is the time in years.

We are given that the simple interest is equal to the principal, which means SI = P.

The annual rate of interest R is given as 16 ⅔% = 50/3% per annum.

Substituting these values into the formula:

P = (P × (50/3) × T) / 100

Dividing both sides by P:

1 = (50 × T) / 300

1 = T / 6

T = 6

This gives the required time period as 6 years.

Explanation

Under compound interest, if a principal amount becomes k times itself in t years, it will become kⁿ times itself in n × t years.

Here, the sum of money doubles (k = 2) in t = 15 years. We want the sum to become 8 times itself.

Since 8 = 2³, we have n = 3.

Substituting the values into the relation, the time required to become 8 times itself is n × t = 3 × 15 = 45 years.

Explanation

Using the depreciation formula: A = P × (1 - r/100)ⁿ

Where: A is the present value = Rs. 4,11,450 r is the rate of decrease = 5% n is the number of years = 3 P is the value of the property 3 years ago

Substituting the given values: 4,11,450 = P × (1 - 5/100)³ 4,11,450 = P × (19/20)³ 4,11,450 = P × (6859/8000)

Rearranging the equation to solve for P: P = 4,11,450 × 8000 / 6859 P ≈ 4,79,895

This rounds to Rs. 4,80,000.

Explanation

Upstream speed is calculated by dividing the upstream distance by the time taken: Upstream speed = Distance / Time = 45 / 6 = 7.5 km/hr

The upstream speed is also defined as the difference between the speed of the boat in still water and the speed of the stream: Upstream speed = Speed of boat - Speed of stream

Let the speed of the stream be v km/hr. Substituting the given values: 7.5 = 10 - v v = 10 - 7.5 = 2.5 km/hr