A number when divided by 7 gives remainder 4. When the same number is divided by 11, it gives remainder 6. What is the number if it lies between 1 and 100?

The correct answer is B: 39. We need to find a number that satisfies two remainder conditions simultaneously. This is a **Chinese Remainder Theorem** problem. **Step 1: Express the first condition** When the number is divided by 7, remainder is 4: \[n = 7k + 4\] where \(k\) is a non-negative integer. This means \(n \in \{4, 11,

A and B working together can complete a job in 6 days. B and C working together can complete it in 9 days. A and C working together can complete it in 12 days. In how many days can all three complete it together?

The correct answer is B: 7.2 days. 1/A + 1/B = 1/6. 1/B + 1/C = 1/9. 1/A + 1/C = 1/12. Adding: 2(1/A + 1/B + 1/C) = 1/6 + 1/9 + 1/12 = 6/36 + 4/36 + 3/36 = 13/36. 1/A + 1/B + 1/C = 13/72. Time = 72/13 ≈ 5.54 days. (Recalculating: 1/6 + 1/9 + 1/12. LCM=36. 6/36 + 4/36 + 3/36 = 13/36. So 2(1/A+1/B+1/C) = 13/36. Combined = 13/72. But th

X can do a work in 12 days, Y can do it in 18 days. They work together for 3 days, then X leaves. In how many more days will Y complete the work?

The correct answer is A: 10.5 days. Work done in 3 days = 3(1/12 + 1/18) = 3(3/36 + 2/36) = 3 × 5/36 = 5/12. Remaining = 7/12. Y completes 7/12 in (7/12)/(1/18) = 7/12 × 18 = 10.5 days

A number when increased by 20% becomes 480. What is the original number?

The correct answer is C: 400. Let original number = x. x × 1.20 = 480. x = 480/1.20 = 400

What is the HCF of 18 and 27?

The correct answer is B: 9. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 27: 1, 3, 9, 27. Common factors: 1, 3, 9. Highest common factor = 9.

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Q.281 Easy HCF and LCM

A man's salary increases from ₹40,000 to ₹48,000. What is the percentage increase?

A15%

B18%

C20%

D22%

Correct Answer: C. 20%

Explanation:

Increase = 48000 - 40000 = 8000. Percentage = 8000/40000 × 100 = 20%.

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Q.282 Easy HCF and LCM

A man walks 5 km/h and covers a distance in 6 hours. If he walks at 6 km/h, how much time saved?

A0.5 hour

B1 hour

C1.5 hours

D2 hours

Correct Answer: B. 1 hour

Explanation:

Distance = 5×6 = 30 km. New time = 30/6 = 5 hours. Time saved = 6-5 = 1 hour.

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Q.283 Easy HCF and LCM

HCF of two numbers is 16. Their LCM is 960. Find the product of the numbers.

A15,360

B16,384

C18,432

D20,480

Correct Answer: A. 15,360

Explanation:

HCF×LCM = product of numbers. 16×960 = 15,360.

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Q.284 Easy HCF and LCM

The HCF of two numbers is 18 and their LCM is 1080. If one number is 108, find the other number.

A180

B162

C216

D144

Correct Answer: A. 180

Explanation:

Using HCF × LCM = Product of two numbers: 18 × 1080 = 108 × x; x = 19440/108 = 180

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Q.285 Easy HCF and LCM

Three bells ring at intervals of 6, 9, and 15 minutes respectively. They ring together at 9:00 AM. At what time will they ring together again?

A9:45 AM

B10:00 AM

C9:30 AM

D10:30 AM

Correct Answer: D. 10:30 AM

Explanation:

To find when the three bells ring together again, we need to find the Least Common Multiple (LCM) of their ringing intervals.

Step 1: Find the prime factorization of each interval

6 = 2 \times 3

\[9 = 3^2\]

15 = 3 \times 5

Step 2: Determine the LCM

The LCM is found by taking the highest power of each prime factor:

\text{LCM}(6, 9, 15) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90

Step 3: Add 90 minutes to the initial time

The bells ring together at 9:00 AM. They will ring together again after 90 minutes.

90 \text{ minutes} = 1 \text{ hour and } 30 \text{ minutes}

Step 4: Calculate the final time

\text{9:00 AM} + 1\text{ hour }30\text{ minutes} = 10:30\text{ AM}

Answer: The three bells will ring together again at 10:30 AM (Option D)

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Q.286 Easy HCF and LCM

A shopkeeper buys books at ₹20 each and sells at a profit of 25%. During a sale, he gives a 10% discount. What is his effective profit percentage?

A12.5%

B15%

C11.25%

D13.75%

Correct Answer: A. 12.5%

Explanation:

To find the effective profit percentage, we need to track the cost price, marked selling price (with profit), and discounted selling price.

Step 1: Calculate the Marked Selling Price (with 25% profit)

The shopkeeper marks up books at a 25% profit on the cost price of ₹20.

\text{Marked Selling Price} = \text{Cost Price} + 25\% \text{ of Cost Price}

= 20 + 0.25 \times 20 = 20 + 5 = ₹25

Step 2: Calculate the Actual Selling Price (after 10% discount)

During the sale, a 10% discount is given on the marked price of ₹25.

\text{Discount} = 10\% \text{ of } 25 = 0.10 \times 25 = ₹2.50

\text{Actual Selling Price} = 25 - 2.50 = ₹22.50

Step 3: Calculate the Effective Profit

\text{Profit} = \text{Actual Selling Price} - \text{Cost Price}

\[= 22.50 - 20 = ₹2.50\]

Step 4: Calculate Effective Profit Percentage

\text{Profit\%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100

= \frac{2.50}{20} \times 100 = \frac{250}{20} = 12.5\%

Answer: The effective profit percentage is 12.5% (Option A)

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Q.287 Easy HCF and LCM

Find the LCM of 48, 64, and 96 using prime factorization.

A192

B288

C384

D576

Correct Answer: A. 192

Explanation:

To find the LCM of 48, 64, and 96, we express each number as a product of prime factors, then take the highest power of each prime that appears.

Step 1: Prime factorization of each number

Divide each number by its prime factors:

48 = 2^4 \times 3^1

\[64 = 2^6\]

96 = 2^5 \times 3^1

Step 2: Identify all prime factors

The prime factors present are: \(2\) and \(3\)

Step 3: Take the highest power of each prime

- Highest power of \(2\): \(2^6\) (from 64)

- Highest power of \(3\): \(3^1\) (from 48 and 96)

Step 4: Calculate the LCM

\text{LCM}(48, 64, 96) = 2^6 \times 3^1 = 64 \times 3 = 192

Answer: The LCM of 48, 64, and 96 is \(192\) (Option A)

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Q.288 Easy HCF and LCM

A worker can complete a job in 12 days. Another worker can complete it in 18 days. If they work together, how many days will they take?

A6.5 days

B7.2 days

C8 days

D9 days

Correct Answer: B. 7.2 days

Explanation:

Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36; Time = 36/5 = 7.2 days

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Q.289 Easy HCF and LCM

A merchant offers successive discounts of 15% and 10%. What is the equivalent single discount?

A23.5%

B25%

C24%

D26.5%

Correct Answer: A. 23.5%

Explanation:

After 15% discount: 85%; After 10% on that: 85×90/100 = 76.5%; Single discount = 100-76.5 = 23.5%

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Q.290 Easy HCF and LCM

If a profit of ₹200 is made on a book by selling at 20% profit, what is the selling price?

A₹1000

B₹1200

C₹1400

D₹1500

Correct Answer: B. ₹1200

Explanation:

Profit = 20% of CP; ₹200 = 0.2×CP; CP = ₹1000; SP = 1000 + 200 = ₹1200

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