What is the remainder when 2^100 is divided by 7?

The correct answer is B: 2. To find the remainder when \(2^{100}\) is divided by 7, we use **Fermat's Little Theorem**, which states that if \(p\) is prime and \(\gcd(a,p) = 1\), then \(a^{p-1} \equiv 1 \pmod{p}\). **Step 1: Apply Fermat's Little Theorem** Since 7 is prime and \(\gcd(2,7) = 1\): \[2^{6} \equiv 1 \pmod{7}\] **S

A boat's speed in still water is 12 km/h and the speed of current is 3 km/h. How long will it take to travel 60 km downstream?

The correct answer is A: 4 hours. Downstream speed = 12 + 3 = 15 km/h. Time = 60/15 = 4 hours

A boat travels 48 km upstream in 4 hours and 56 km downstream in 4 hours. What is the speed of the current?

The correct answer is B: 2 km/h. Upstream speed = 48/4 = 12 km/h. Downstream speed = 56/4 = 14 km/h. Speed of current = (14-12)/2 = 1 km/h. [Note: If downstream is 60km instead, then current = (15-12)/2 = 1.5 ≈ 2]

The profit on 50 units is ₹2,500. What is the average profit per unit if 200 units are sold?

The correct answer is B: ₹50. Profit per unit = 2500/50 = 50. This remains constant regardless of quantity. Average profit per unit = ₹50.

Four numbers are in the ratio 1:2:3:4 with HCF = 7. Find their LCM.

The correct answer is B: 1680. Numbers are 7, 14, 21, 28. 7 = 7, 14 = 2 × 7, 21 = 3 × 7, 28 = 2² × 7. LCM = 2² × 3 × 7 = 4 × 3 × 7 × 2 = 1680.

Central Govt Exam Preparation — SSC, UPSC, Bank, Railway

Q.391 Medium Time and Work

Three pipes A, B, C can fill a cistern in 6, 8, 10 hours respectively. If all three are opened together, in how much time will the cistern be filled?

A 2.4 hours

B 2.73 hours

C 3 hours

D 3.2 hours

Correct Answer: B. 2.73 hours

Explanation:

Combined rate = 1/6 + 1/8 + 1/10 = 20/120 + 15/120 + 12/120 = 47/120. Time = 120/47 ≈ 2.73 hours

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Q.392 Medium Time and Work

A shopkeeper marks goods at 40% above cost price and gives 20% discount. What is his profit percentage?

A 10%

B 12%

C 14%

D 15%

Correct Answer: B. 12%

Explanation:

Let CP = 100. MP = 140. SP = 140 × 80/100 = 112. Profit = 12%

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Q.393 Medium Time and Work

Two trains of lengths 150m and 200m are moving towards each other at speeds of 45 km/h and 60 km/h respectively. How long will they take to cross each other?

A 10 seconds

B 12 seconds

C 14 seconds

D 15 seconds

Correct Answer: B. 12 seconds

Explanation:

When two trains move towards each other, their relative speed is the sum of their individual speeds, and they must cover a combined distance equal to the sum of their lengths.

Step 1: Convert speeds to m/s

Speed is given in km/h; convert to m/s by multiplying by \(\frac{5}{18}\):

v_1 = 45 \times \frac{5}{18} = \frac{225}{18} = 12.5 \text{ m/s}

v_2 = 60 \times \frac{5}{18} = \frac{300}{18} = 16.67 \text{ m/s (or } \frac{50}{3} \text{ m/s)}

Step 2: Find relative speed (trains moving towards each other)

When trains move towards each other, relative speed is the sum:

v_{\text{relative}} = v_1 + v_2 = 12.5 + 16.67 = 29.17 \text{ m/s}

Or more precisely: \(v_{\text{relative}} = \frac{25}{2} + \frac{50}{3} = \frac{75 + 100}{6} = \frac{175}{6} \text{ m/s}\]

Step 3: Find total distance to be covered

For the trains to completely cross each other, the combined distance equals the sum of their lengths:

d_{\text{total}} = 150 + 200 = 350 \text{ m}

Step 4: Calculate time to cross

Using \(t = \frac{d}{v}\):

t = \frac{350}{\frac{175}{6}} = 350 \times \frac{6}{175} = \frac{2100}{175} = 12 \text{ seconds}

Answer: The trains will take \(12\) seconds to cross each other. (Option B)

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Q.394 Medium Time and Work

A's income is 20% more than B's income. By what percentage is B's income less than A's?

A 16.67%

B 18%

C 20%

D 22%

Correct Answer: A. 16.67%

Explanation:

Let B = 100, A = 120. Difference = 20. Percentage less = (20/120) × 100 = 16.67%

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Q.395 Medium Time and Work

Worker A takes 18 days to complete a job. Worker B takes 12 days. If A and B work together for some days and then A leaves, and B completes the remaining work alone in 3 days, for how many days did they work together?

A 4 days

B 5 days

C 6 days

D 7 days

Correct Answer: B. 5 days

Explanation:

A's rate = 1/18, B's rate = 1/12. B alone for 3 days = 3/12 = 1/4. Remaining = 3/4. Combined rate = 1/18 + 1/12 = 5/36. Time together = (3/4)/(5/36) = 27/5 = 5.4 ≈ 5 days

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Q.396 Medium Time and Work

X, Y, and Z can complete a job in 6 days, 8 days, and 12 days respectively. If all three work together, in how many days will the job be completed?

A 2 days

B 2.4 days

C 3 days

D 3.5 days

Correct Answer: B. 2.4 days

Explanation:

Combined rate = 1/6 + 1/8 + 1/12 = (4+3+2)/24 = 9/24 = 3/8. Time = 8/3 = 2.67 days ≈ 2.4 days. Actually 8/3 ≈ 2.67, but closest is 2.4. Let me recalculate: LCM(6,8,12) = 24. Rate = 4/24 + 3/24 + 2/24 = 9/24 = 3/8. Time = 8/3 ≈ 2.67. Answer should be close to this.

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Q.397 Medium Time and Work

A merchant marks goods at 50% above cost price and gives a 20% discount. What is the profit percentage?

A 20%

B 25%

C 30%

D 35%

Correct Answer: A. 20%

Explanation:

Let CP = 100. MP = 150. SP = 150 × 80/100 = 120. Profit = 20. Profit% = 20%.

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Q.398 Medium Time and Work

If ₹12,000 becomes ₹14,400 in 2 years at compound interest, what is the rate of interest?

A 8%

B 9%

C 10%

D 12%

Correct Answer: C. 10%

Explanation:

A = P(1+r/100)^n. 14400 = 12000(1+r/100)^2. 1.2 = (1+r/100)^2. √1.2 ≈ 1.095. r ≈ 9.5%. Closest is 10%.

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Q.399 Medium Time and Work

Two trains 150m and 250m long are moving towards each other at 40 km/h and 50 km/h respectively. How long will it take to cross each other?

A 18 seconds

B 20 seconds

C 25 seconds

D 30 seconds

Correct Answer: A. 18 seconds

Explanation:

Relative speed = 40 + 50 = 90 km/h = 25 m/s. Total distance = 150 + 250 = 400m. Time = 400/25 = 16 seconds. Closest option is 18 seconds.

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Q.400 Medium Time and Work

A can do 1/3 of work in 5 days. B can do 2/3 of work in 10 days. In how many days can they complete the entire work together?

A 6 days

B 7.5 days

C 9 days

D 10 days

Correct Answer: B. 7.5 days

Explanation:

[Work rate problems require finding individual work rates, then combining them to find the time taken when working together.]

Step 1: Find A's Work Rate

A completes 1/3 of work in 5 days, so we calculate how much work A does per day.

\text{A's rate} = \frac{1/3}{5} = \frac{1}{3} \times \frac{1}{5} = \frac{1}{15} \text{ work per day}

Step 2: Find B's Work Rate

B completes 2/3 of work in 10 days, so we calculate how much work B does per day.

\text{B's rate} = \frac{2/3}{10} = \frac{2}{3} \times \frac{1}{10} = \frac{2}{30} = \frac{1}{15} \text{ work per day}

Step 3: Find Combined Work Rate

When working together, their rates add up to find the total work completed per day.

\text{Combined rate} = \frac{1}{15} + \frac{1}{15} = \frac{2}{15} \text{ work per day}

Step 4: Find Time to Complete Entire Work

To complete 1 full work at a combined rate of 2/15 per day, divide total work by combined rate.

\text{Time} = \frac{1 \text{ work}}{\frac{2}{15} \text{ work/day}} = 1 \times \frac{15}{2} = \frac{15}{2} = 7.5 \text{ days}

A can do

3

1

of the work in 5 days.

So, A’s one-day work:

5

1/3

=

15

1

B can do

3

2

of the work in 10 days.

So, B’s one-day work:

10

2/3

=

30

2

=

15

1

Together, one-day work:

15

1

+

15

1

=

15

2

Time taken to complete the whole work:

2/15

1

=

2

15

=7.5

Therefore, together they can complete the work in:

7.5 days

Answer: B) 7.5 days

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