RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

In a proportion a:b :: c:d, the product of extremes equals the product of means (a × d = b × c).

For the first proportion, 4 × x = 28 × 4.1 x = (28 × 4.1) / 4 = 7 × 4.1 = 28.7

For the second proportion, 4 × y = 32 × 3 y = (32 × 3) / 4 = 8 × 3 = 24

The ratio of x to y is x : y = 28.7 : 24 = 287 : 240.

Explanation

The profit sharing ratio is directly proportional to the investment ratio when the time period is same for both investors.

Total ratio parts = 31 + 34 = 65 parts Difference in ratio parts = 34 - 31 = 3 parts

Difference in profit shares = Total Profit × (Difference in parts / Total parts) Difference = 44,850 × (3 / 65) = 690 × 3 = ₹2,070

Explanation

Since AB is the diameter, the angle subtended by AB at any point on the circle is a right angle, so ∠ACB = 90° and ∠ADB = 90°.

In right-angled triangle ABC, ∠ABC = 90° - ∠CAB = 90° - 48° = 42°. In right-angled triangle ABD, ∠BAD = 90° - ∠ABD = 90° - 55° = 35°.

Since C and D lie on opposite sides of the diameter AB: ∠CBD = ∠ABC + ∠ABD = 42° + 55° = 97° ∠CAD = ∠CAB + ∠BAD = 48° + 35° = 83°

Difference = ∠CBD - ∠CAD = 97° - 83° = 14°.

Shortcut formula: Difference = 2 × (∠ABD - ∠CAB) = 2 × (55° - 48°) = 14°.

Explanation

Using the inverse variation formula M₁ × D₁ = M₂ × D₂:

Initial number of men (M₁) = 100 Initial number of days (D₁) = 27 Remaining men (M₂) = 100 - 55 = 45

Substituting the values into the formula: 100 × 27 = 45 × D₂ D₂ = (100 × 27) / 45 D₂ = 2700 / 45 = 60 days

Explanation

Let the ratio multiplier be x. The number of articles of types P, Q, and R purchased are 4x, 2x, and 5x, respectively.

Total cost = (4x × ₹280) + (2x × ₹180) + (5x × ₹115) = ₹6,165

1120x + 360x + 575x = 6165

2055x = 6165

x = 6165 ÷ 2055 = 3

Number of articles of type Q = 2x = 2 × 3 = 6.

Explanation

The modal class is the class interval with the highest frequency, which is 30-40 (frequency = 20).

The formula for the mode of grouped data is: Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Where: L = lower limit of modal class = 30 f₁ = frequency of modal class = 20 f₀ = frequency of preceding class = 12 f₂ = frequency of succeeding class = 10 h = class width = 10

Substituting the values into the formula: Mode = 30 + [(20 - 12) / (2 × 20 - 12 - 10)] × 10 Mode = 30 + [8 / (40 - 22)] × 10 Mode = 30 + (8 / 18) × 10 Mode = 30 + 4.44 = 34.44

Explanation

Average Speed = Total Distance / Total Time

Total Distance = (20 m/min × 90 min) + (55 m/min × 60 min) = 1800 m + 3300 m = 5100 m

Total Time = 90 min + 60 min = 150 min

Average Speed = 5100 / 150 = 34 m/min

Explanation

Average speed = Total distance / Total time

The distances of segments PQ, QR, and RS are in the ratio 2:3:1 with a total distance of 480 km. Sum of ratio parts = 2 + 3 + 1 = 6 parts. Distance of PQ = (2/6) × 480 = 160 km Distance of QR = (3/6) × 480 = 240 km Distance of RS = (1/6) × 480 = 80 km

Time taken for each segment: Time for PQ = 160 / 48 = 10/3 hours Time for QR = 240 / 72 = 10/3 hours Time for RS = 80 / 96 = 5/6 hours

Total time = 10/3 + 10/3 + 5/6 = 45/6 = 15/2 hours

Average speed = 480 / (15/2) = 480 × 2 / 15 = 64 km/h

Explanation

Total surface area of a solid cube = 6 × a², where a is the side length.

Given side length a = 84 cm.

a² = 84² = 7056 cm²

Total surface area = 6 × 7056 = 42336 cm²

Explanation

Under simple interest, when a sum of money P becomes 5 times itself in time T = 5 years, the total amount A = 5P.

Simple Interest (SI) = A - P = 5P - P = 4P.

Using the formula SI = (P × R × T) / 100: 4P = (P × R × 5) / 100 4 = 5R / 100 R = 400 / 5 = 80%

Explanation

Using the compound interest amount formula A = P(1 + r/100)ᵗ, where r = 10%:

Piyush's Amount = Manoj's Amount P × (1 + 10/100)¹⁸ = M × (1 + 10/100)¹⁹ P × (1.1)¹⁸ = M × (1.1)¹⁹ P = M × 1.1 = 11M/10

The ratio of Piyush's share to Manoj's share is P : M = 11 : 10. Total parts = 11 + 10 = 21 parts = ₹1407. 1 part = 1407 / 21 = ₹67.

Manoj's share = 10 parts = 10 × 67 = ₹670.

Explanation

Let N be the total number of registered voters.

Polled votes = 0.80 × N Valid votes = 0.90 × (0.80 × N) = 0.72 × N

The winning candidate received 89% of the valid votes, leaving 11% for the losing candidate. The winning margin is 89% − 11% = 78% of the valid votes.

Margin = 0.78 × 0.72 × N = 0.5616 × N

Given margin = 351: 0.5616 × N = 351 N = 351 / 0.5616 = 625

Explanation

Work done by (A + B + C) in 1 day = 1/54 Work done by (B + C) in 1 day = 1/56

Work done by A in 1 day = 1/54 - 1/56 = 2 / (54 × 56) = 1/1512

Since B is 75% more efficient than A: Work done by B in 1 day = 1.75 × (1/1512) = (7/4) × (1/1512) = 1/864

Work completed by A, B, and C together in 52 days = 52/54 = 26/27 Remaining work = 1 - 26/27 = 1/27

Time taken by B alone to complete the remaining work = (1/27) ÷ (1/864) = 864 ÷ 27 = 32 days.

Explanation

Evaluate the cube root of each decimal term individually:

∛0.216 = 0.6 ∛0.001 = 0.1 ∛0.008 = 0.2

Substitute these values back into the expression:

0.6 + 0.1 - 0.2 = 0.5

Explanation

Using the algebraic identity (x - y)² = x² + y² - 2xy, we can express x² + y² + xy as:

x² + y² + xy = (x - y)² + 3xy

Substituting the given values x - y = 5 and xy = 21:

x² + y² + xy = 5² + 3 × 21 = 25 + 63 = 88

Explanation

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

In the number 718A5104, the last two digits form 04, which is divisible by 4 (04 ÷ 4 = 1). Since divisibility by 4 depends solely on the last two digits, the value of digit A has no effect on the divisibility of the entire number. Therefore, A can take any single-digit value from 0 to 9.

The maximum possible single-digit value for A is 9.

Explanation

Radius of the circular pipe r = 1 cm, flow rate v = 5 m/s = 500 cm/s, radius of the tank R = 100 cm, and time t = 13 minutes = 780 seconds.

Volume of water discharged by pipe = Volume of water added to tank π × r² × v × t = π × R² × h

Substituting the given values: 1² × 500 × 780 = 100² × h 390,000 = 10,000 × h h = 39 cm

Explanation

Cost Price per dozen = ₹630

Selling Price per dozen = 12 × ₹63 = ₹756

Profit per dozen = Selling Price - Cost Price = ₹756 - ₹630 = ₹126

Profit Percentage = (Profit / Cost Price) × 100 = (126 / 630) × 100 = 20%

Since profit percentage depends entirely on the ratio of unit cost price to unit selling price, the total quantity of 42 dozens does not affect the calculation.

Explanation

Raghav's monthly expenditure on house rent, food, and utility bills = ₹7,500 + ₹3,500 = ₹11,000.

Monthly savings = Monthly salary - Monthly expenditure = ₹24,000 - ₹11,000 = ₹13,000.

Since he spends his entire savings in his birthday month, he saves money during 11 months of the year.

Total annual savings = ₹13,000 × 11 = ₹1,43,000.

Explanation

The mean of grouped data is given by the formula Mean = Σ(f × x) / Σf, where f represents the class frequency and x represents the mid-point (class mark) of each class interval.

The mid-points for the class intervals are: For 0-8: x = 4 For 8-16: x = 12 For 16-24: x = 20 For 24-32: x = 28 For 32-40: x = 36 For 40-48: x = 44

Sum of frequencies Σf = 2 + 4 + 7 + 5 + 3 + 6 = 27

Sum of products Σ(f × x) = (2 × 4) + (4 × 12) + (7 × 20) + (5 × 28) + (3 × 36) + (6 × 44) = 8 + 48 + 140 + 140 + 108 + 264 = 708

Mean = 708 / 27 = 26.222... ≈ 26.22

Explanation

Average speed is calculated using the formula: Average Speed = Total Distance ÷ Total Time.

Calculating the time taken for each part of the journey: Time 1 = 48 km ÷ 12 km/h = 4 hours Time 2 = 52 km ÷ 13 km/h = 4 hours Time 3 = 120 km ÷ 15 km/h = 8 hours

Total Distance = 48 + 52 + 120 = 220 km Total Time = 4 + 4 + 8 = 16 hours

Average Speed = 220 km ÷ 16 hours = 13.75 km/h

Explanation

Let the cost price of 1000 gm of goods be ₹1000.

Since the shopkeeper claims to sell at a 55% loss, the selling price charged for the fake weight is ₹1000 × (1 - 0.55) = ₹450.

Let the actual weight used by him be x gm. The actual cost price of x gm of goods to him is ₹x.

Since he actually gains 75% profit on this transaction: Selling Price = Cost Price × (1 + Profit%) 450 = x × (1 + 0.75) 450 = 1.75 × x x = 450 ÷ 1.75 = 1800 ÷ 7 = 257.14 gm ≈ 257.1 gm

Explanation

Profit share is proportional to the product of capital invested and the time period of investment.

Let the initial monthly investments of X, Y, and Z be 5, 3, and 7 units respectively.

Effective capital-month investment for 1 year: X = (5 × 6) + (2.5 × 6) = 45 units Y = 3 × 12 = 36 units Z = 7 × 12 = 84 units

Ratio of profit shares X : Y : Z = 45 : 36 : 84 = 15 : 12 : 28

Total ratio units = 15 + 12 + 28 = 55 units

Value per ratio unit = ₹55,000 / 55 = ₹1,000

Difference between X's and Y's profit share = (15 - 12) × ₹1,000 = 3 × ₹1,000 = ₹3,000

Explanation

Using the compound interest formula A = P(1 + r/100)ⁿ, where P is the principal, r is the annual rate, and n is the time in years.

Let Krish's share be K and Veer's share be V. Since r = 10%, the annual multiplier is (1 + 10/100) = 11/10.

Equating Krish's amount after 13 years to Veer's amount after 14 years: K × (11/10)¹³ = V × (11/10)¹⁴

Dividing both sides by (11/10)¹³: K = V × (11/10) K / V = 11 / 10

The ratio of Krish's share to Veer's share is 11 : 10. The sum of ratio parts is 11 + 10 = 21 parts.

Value of 1 part = 1407 / 21 = 67

Veer's share = 10 × 67 = ₹670

Explanation

Following the BODMAS rule (Order of Operations), perform division and multiplication from left to right before addition and subtraction:

Division: (-56) ÷ (-14) = 4 Multiplication: (-2) × 8 = -16

Substituting these back into the expression gives: (-9) - 4 + (-16)

Now, perform subtraction and addition from left to right: -9 - 4 = -13 -13 - 16 = -29

Explanation

For a sum of money to double itself under simple interest, the rate of interest per annum is given by the formula:

R = [(n - 1) × 100] / T

Here, n = 2 (since the sum doubles) and T = 15 years.

R = [(2 - 1) × 100] / 15 = 100 / 15 = 20 / 3 = 6.666... ≈ 6.67%

Explanation

Upstream speed = Distance / Upstream time = 80 / 4 = 20 km/h.

Speed of current = Speed in still water - Upstream speed = 25 - 20 = 5 km/h.

Downstream speed = Speed in still water + Speed of current = 25 + 5 = 30 km/h.

Downstream time = Distance / Downstream speed = 80 / 30 = 2⅔ hours = 2 hours 40 minutes.

Total time = Upstream time + Downstream time = 4 hours + 2 hours 40 minutes = 6 hours 40 minutes.

Explanation

The bells ring together at time intervals equal to the Least Common Multiple (LCM) of their individual ringing times.

LCM(18, 24, 30) = 360 minutes = 6 hours.

The bells ring together every 6 hours.

1st time = 6:30 AM 2nd time = 6:30 AM + 6 hours = 12:30 PM 3rd time = 12:30 PM + 6 hours = 6:30 PM

Explanation

Let the two consecutive odd natural numbers be n - 1 and n + 1, so their sum is 2n.

Using the algebraic expansion (n - 1)² + (n + 1)² = 2n² + 2, substitute the given sum of squares:

2n² + 2 = 5002 2n² = 5000 n² = 2500 n = 50

The sum of the two numbers is 2n = 2 × 50 = 100.

Explanation

Let Cost Price (CP) = 100

Marked Price (MP) = 100 + 150 = 250

Selling Price (SP) to gain 130% = 100 + 130 = 230

Discount = MP - SP = 250 - 230 = 20

Discount % = (Discount / MP) × 100 = (20 / 250) × 100 = 8%

Explanation

The overall average is calculated using the weighted average formula: Overall Average = Σ(n × x) / Σn.

Simplifying the ratio of group members for groups A, B, and C: Ratio = 30 : 72 : 75 = 10 : 24 : 25 Sum of ratio weights = 10 + 24 + 25 = 59

Weighted total expenditure = (10 × 434) + (24 × 324) + (25 × 280) = 4340 + 7776 + 7000 = 19116

Overall average = 19116 / 59 = 324

Explanation

Substituting x = 2, y = -2, and z = 5 into the three equations verifies the solution directly:

First equation: 3(2) - 4(-2) = 6 + 8 = 14, and 6(5) - 16 = 30 - 16 = 14.

Second equation: -4(2) + (-2) + 5 = -8 - 2 + 5 = -5.

Third equation: 3(-2) + 2(5 - 1) = -6 + 8 = 2, which equals x.

All three equations are satisfied simultaneously.

Explanation

The mean deviation about the mean is given by the formula Σ|x - x̄| / n, where x̄ is the arithmetic mean and n is the total number of observations.

First, calculate the mean: x̄ = (58 + 65 + 42 + 66 + 49 + 59 + 44 + 45 + 62 + 60) / 10 = 550 / 10 = 55

Next, find the absolute deviations |x - x̄| from the mean: |58 - 55| = 3, |65 - 55| = 10, |42 - 55| = 13, |66 - 55| = 11, |49 - 55| = 6, |59 - 55| = 4, |44 - 55| = 11, |45 - 55| = 10, |62 - 55| = 7, |60 - 55| = 5

Sum of absolute deviations = 3 + 10 + 13 + 11 + 6 + 4 + 11 + 10 + 7 + 5 = 80

Mean deviation = 80 / 10 = 8

Explanation

Using the work equivalence formula: M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Here, M₁ = 15 men, D₁ = 20 days, H₁ = 6 hours per day, M₂ = 18 men, H₂ = 5 hours per day, and D₂ is the required number of days.

Substituting the values into the formula: 15 × 20 × 6 = 18 × D₂ × 5 1800 = 90 × D₂ D₂ = 1800 / 90 = 20 days

Explanation

The line segment AO joining external point A to the centre O perpendicularly bisects the chord BC at point M.

BM = BC ÷ 2 = 24 ÷ 2 = 12 cm

In right-angled triangle AMB: AM = √(AB² - BM²) = √(30² - 12²) = √(900 - 144) = √756 = 6√21 cm

In right-angled triangle ABO, ∠ABO = 90° and BM is the altitude to hypotenuse AO. By similar triangles: BM² = AM × MO 12² = 6√21 × MO 144 = 6√21 × MO MO = 24 / √21 cm

In right-angled triangle BMO: r² = BM² + MO² = 12² + (24 / √21)² = 144 + 576 / 21 = 144 + 192 / 7 = 1200 / 7 ≈ 171.43 r = √(1200 / 7) ≈ 13.09 cm

Rounding off to the nearest integer yields 13 cm.

Explanation

If C lies on the minor arc AB, the angle subtended by chord AB at any point D on the major arc is ∠ADB = 180° - ∠ACB = 180° - 124° = 56°.

The central angle subtended by chord AB at the center P is twice the angle at the major arc: ∠APB = 2 × ∠ADB = 2 × 56° = 112°

In quadrilateral APBM, since radii PA and PB are perpendicular to tangents MA and MB, the opposite angles ∠APB and ∠AMB are supplementary: ∠AMB = 180° - ∠APB = 180° - 112° = 68°

Direct formula: ∠AMB = 2 × ∠ACB - 180° = 2(124°) - 180° = 68°

Explanation

Ratio of sugar to flour = 3 : 4

Using direct proportion: Sugar / Flour = 3 / 4

Substituting 12 cups for sugar: 12 / Flour = 3 / 4 Flour = 12 × (4 / 3) = 16 cups

Explanation

For six consecutive even numbers, the average equals the middle value between the third and fourth numbers. Expressing the six numbers as x, x + 2, x + 4, x + 6, x + 8, and x + 10, their average is x + 5.

Given average = 175: x + 5 = 175 x = 170

The smallest of these numbers is 170.

Explanation

The formula for the volume of a hemisphere is V = (2/3)πr³, where r is the radius. Given the diameter is 14 cm, the radius r = 7 cm. Substituting r = 7 cm and π = 22/7 into the formula:

V = (2/3) × (22/7) × 7³ V = (2/3) × 22 × 49 V = 2156 / 3 = 718.67 cm³

Explanation

Net filling rate per hour when all three taps are opened together = 1/4 + 1/37 - 1/37 = 1/4 of the cistern per hour.

To fill one-fourth of the empty cistern: Time = (1/4) ÷ (1/4) = 1 hour.

Explanation

To find the mode of a grouped frequency distribution with inclusive class boundaries, first convert the classes into continuous boundaries by adjusting each limit by 0.5:

34.5 - 44.5: 10 44.5 - 54.5: 20 54.5 - 64.5: 30 64.5 - 74.5: 25 74.5 - 84.5: 10 84.5 - 94.5: 5

The modal class is 54.5 - 64.5 as it has the maximum frequency of 30.

Mode formula: Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Where: L = lower limit of modal class = 54.5 f₁ = frequency of modal class = 30 f₀ = frequency of preceding class = 20 f₂ = frequency of succeeding class = 25 h = class size = 10

Mode = 54.5 + [(30 - 20) / (2 × 30 - 20 - 25)] × 10 Mode = 54.5 + [10 / (60 - 45)] × 10 Mode = 54.5 + (10 / 15) × 10 Mode = 54.5 + 6.67 = 61.17

Explanation

Average Speed = Total Distance / Total Time

Time taken for segment 1 = 336 / 24 = 14 hours Time taken for segment 2 = 925 / 5 = 185 hours Time taken for segment 3 = 432 / 12 = 36 hours

Total Distance = 336 + 925 + 432 = 1693 km Total Time = 14 + 185 + 36 = 235 hours

Average Speed = 1693 / 235 = 7 48/235 km/hr

Explanation

The length of an arc of a sector with central angle θ and radius r is given by the formula L = (θ / 360°) × 2πr.

Substituting θ = 45°, r = 8 units, and π = 22/7:

L = (45° / 360°) × 2 × (22 / 7) × 8

Since 45° / 360° = 1/8:

L = (1 / 8) × 2 × (22 / 7) × 8 = 2 × (22 / 7) = 44 / 7 units

Explanation

Let the required number be x.

Since 20% of x exceeds 20% of 620 by 220: 20% of (x - 620) = 220 (1 / 5) × (x - 620) = 220 x - 620 = 220 × 5 x - 620 = 1100 x = 1100 + 620 = 1720

Explanation

Let the number of chocolates purchased of each type be 9.

Cost price of 9 chocolates of the first type = ₹8 Cost price of 9 chocolates of the second type = 9 × ₹8 = ₹72 Total cost price of 18 chocolates = ₹8 + ₹72 = ₹80

Selling price of 1 chocolate = ₹8 Total selling price of 18 chocolates = 18 × ₹8 = ₹144

Profit = Total selling price - Total cost price = ₹144 - ₹80 = ₹64 Profit percentage = (Profit / Total cost price) × 100 = (64 / 80) × 100 = 80%

Explanation

Net filling rate per hour when all three taps are opened together = 1/4 + 1/31 - 1/31 = 1/4 of the cistern per hour.

Time required to fill 1/4 of the cistern = (1/4) ÷ (1/4) = 1 hour.

Explanation

To find the minimum value of k, we determine the smallest valid values for a, b, and c.

Since 12 divides b and 18 divides b, b must be a multiple of the least common multiple of 12 and 18: lcm(12, 18) = 36, so b ≥ 36.

Since gcd(a, b) = 12, 12 divides a, so a ≥ 12.

Since gcd(b, c) = 18, 18 divides c, so c ≥ 18.

Testing the smallest possible values a = 12, b = 36, and c = 18: gcd(12, 36) = 12 gcd(36, 18) = 18 gcd(18, 12) = 6

All conditions are satisfied. The minimum sum is: a + b + c = 12 + 36 + 18 = 66

Using a + b + c = 6k: 66 = 6k k = 11

Explanation

Sum of numbers = Average × Count

Sum of the 5 original numbers = 5 × 20 = 100 Sum of the 4 remaining numbers = 4 × 18 = 72

Excluded number = 100 − 72 = 28

Explanation

The mean proportional between two numbers a and b is given by √(a × b).

First, calculate the initial mean proportional between 18 and 32: √(18 × 32) = √576 = 24

Increasing this value by 25%: 24 × 1.25 = 30

Given that 30 is the mean proportional between 18 and y: √(18 × y) = 30 18 × y = 30² = 900 y = 900 / 18 = 50

Explanation

Expanding the left side of (x - 1)(x + 2) = x² + 3 yields x² + x - 2 = x² + 3. Subtracting x² from both sides simplifies the equation to x - 5 = 0. Since the x² terms cancel out, the degree of the equation is 1, making it a linear equation rather than a quadratic equation.