Clerkship Arithmetic Quiz: 15 Topics, 15 Questions
A little practice. A step forward.
One question from each arithmetic topic of the Clerkship 2024 syllabus, with an explanation for every answer. 📖 Read the detailed topic
If the number 47x2 is divisible by 9, what is x?
A number is divisible by 9 when its digit sum is a multiple of 9. 4 + 7 + x + 2 = 13 + x, and 13 + x = 18 gives x = 5. For x = 4 the sum is 17 and for x = 6 it is 19, neither a multiple of 9.
- For 3 and 9, look at the digit sum
- The number is divisible by 9 exactly when its digit sum is
- For 3 the digit sum only has to be a multiple of 3; for 9 it has to be a multiple of 9
- Lift the sum to the next multiple of 9: from 13 to 18, a gap of 5
| Divisor | Rule |
|---|---|
| 3 | Digit sum is a multiple of 3 |
| 4 | Number formed by the last two digits is a multiple of 4 |
| 9 | Digit sum is a multiple of 9 |
| 11 | Difference of the sums of digits in odd and even places is 0 or a multiple of 11 |
What is the value of $\frac{3}{8} + \frac{5}{12}$?
The LCM of 8 and 12 is 24. $\frac{3}{8} = \frac{9}{24}$ and $\frac{5}{12} = \frac{10}{24}$, so the sum is $\frac{19}{24}$. $\frac{8}{20}$ comes from adding numerators and denominators separately, which is wrong.
- Make the denominators equal before adding
- Using the LCM keeps the numbers small
- Writing $\frac{8}{20}$ by adding tops and bottoms separately
- Denominators are never added; only numerators are
What is 0.6 × 0.07?
Ignore the decimal points: 6 × 7 = 42. 0.6 has 1 decimal place and 0.07 has 2, so 3 in all. Moving the point 3 places from the right gives 0.042. A quick estimate agrees: 60% of 0.07 is about 0.04.
- In multiplication, the decimal places add
- Estimate to check where the point goes
- Miscounting the places and writing 0.42 or 0.0042
- Count the places and move the point from the right
What is $0.\overline{36}$ as a fraction in lowest terms?
By the rule for a two-digit repeating decimal, $0.\overline{36} = \frac{36}{99}$. Dividing both by 9 gives $\frac{4}{11}$. $\frac{9}{25} = 0.36$ is a terminating decimal, not a repeating one.
- A two-digit repeating block gives the denominator 99
- Simplify the fraction you get
- Treating $0.\overline{36}$ and 0.36 as the same (the latter is $\frac{9}{25}$)
- One 9 in the denominator for each repeating digit
| Repeating decimal | Fraction | Lowest terms |
|---|---|---|
| $0.\overline{3}$ | $\frac{3}{9}$ | $\frac{1}{3}$ |
| $0.\overline{36}$ | $\frac{36}{99}$ | $\frac{4}{11}$ |
| $0.1\overline{6}$ | $\frac{16 - 1}{90}$ | $\frac{1}{6}$ |
| $0.\overline{9}$ | $\frac{9}{9}$ | 1 |
What is the value of 48 ÷ 6 × 2 + (15 − 3 × 4)?
Brackets first: 15 − 3 × 4 = 15 − 12 = 3. Then left to right: 48 ÷ 6 = 8 and 8 × 2 = 16. Finally 16 + 3 = 19. The answer 7 comes from reading 48 ÷ (6 × 2) = 4; 16 comes from leaving out the bracket.
- Brackets come first in BODMAS
- Division and multiplication rank equally, so go left to right
- Reading 48 ÷ 6 × 2 as 48 ÷ (6 × 2)
- Brackets, then ÷ × from the left, then + −
What is the HCF of 84 and 126?
$84 = 2^{2} \times 3 \times 7$ and $126 = 2 \times 3^{2} \times 7$. Taking the smallest power of each common factor gives $2 \times 3 \times 7 = 42$. 21 divides both but is not the greatest; 63 does not divide 84.
- For the HCF take the smallest powers
- HCF × LCM = product of the two numbers
- Finding a common factor but forgetting to find the greatest one
- The HCF is small (common part), the LCM is big
| HCF | LCM | |
|---|---|---|
| Meaning | Greatest common factor | Least common multiple |
| In prime factors | Smallest powers | Largest powers |
| Typical problem | Largest length or tile | When together again |
| Compared with the numbers | Equal or smaller | Equal or larger |
What is the smallest number that leaves remainder 3 when divided by 4, 5 and 6?
The LCM of 4, 5 and 6 is 60, which they all divide exactly. To leave remainder 3 each time, add 3: 60 + 3 = 63. 60 itself leaves remainder 0; 123 also fits, but the smallest is 63.
- When the remainder is the same, answer = LCM + remainder
- Writing the LCM 60 as the answer
- Find the LCM, then add the remainder
A invests ₹6,000 for 12 months and B invests ₹8,000 for 6 months. If the profit at the end of the year is ₹7,500, what is A's share?
Profit is shared in the ratio of capital × time: A : B = 6000 × 12 : 8000 × 6 = 72000 : 48000 = 3 : 2. A's share is 7500 ÷ 5 × 3 = ₹4,500. ₹3,750 would be an equal split.
- Profit share = ratio of capital × time
- Keep the time in the same unit
- Ignoring time and using only 6 : 8
- Rupees × months = 'rupee-months'
The average of 5 numbers is 20. If a sixth number 32 is added, what is the average of the six numbers?
Sum = average × count = 5 × 20 = 100. Adding 32 gives 132. New average = 132 ÷ 6 = 22. The figure 26 comes from (20 + 32) ÷ 2, which is the wrong method.
- Sum = average × count
- Always work with the total sum and the total count
- Averaging the old average with the new number
- Go to the sum first, then come back to the average
If A : B = 2 : 3 and B : C = 4 : 5, what is A : B : C?
B is 3 in one ratio and 4 in the other; their LCM is 12. Multiply the first by 4: A : B = 8 : 12. Multiply the second by 3: B : C = 12 : 15. So A : B : C = 8 : 12 : 15. The option 2 : 3 : 5 skips equalising B.
- Make the common term B equal using the LCM
- Writing 2 : 3 : 5 without matching B
- Match the middle term
The price of an article is raised by 20% and then reduced by 20%. What is the net change in price?
Take the price as 100. After a 20% rise it is 120. A 20% cut of 120 is 24, leaving 96. The net change is a 4% decrease. By the formula, $20 - 20 + \frac{20 \times (-20)}{100} = -4$. The two percentages have different bases, so they do not cancel.
- The base is always the earlier value
- Take the price as 100 to keep it simple
- Thinking +20 and −20 cancel out
- 100 → 120 → 96
| Step | Price (taking 100) |
|---|---|
| Start | 100 |
| After the 20% rise | 120 |
| After cutting 20% of 120 = 24 | 96 |
| Net change | 4% less |
What is the simple interest on ₹8,000 at 5% per annum for 3 years?
Simple interest $= \frac{P \times R \times T}{100} = \frac{8000 \times 5 \times 3}{100} = 1200$. That is ₹400 a year, so ₹1,200 in three years. The amount would be ₹9,200, not the interest.
- Interest = principal × rate × time ÷ 100
- Simple interest is the same every year
- Mixing up interest and amount
- First the interest for one year, then multiply by the years
An article bought for ₹400 is sold for ₹460. What is the profit percentage?
Profit = 460 − 400 = ₹60. Profit percentage is on the cost price: 60 ÷ 400 × 100 = 15%. 13% comes from wrongly using the selling price 460 as the base, and 60 is the rupee profit, not a percentage.
- Profit and loss percentages use the cost price
- Selling price = cost price + profit
- Dividing by the selling price
- Always divide by the price you bought at
A person goes along a route at 40 km/h and returns along the same route at 60 km/h. What is the average speed for the whole journey?
For equal distances the average speed is $\frac{2xy}{x + y} = \frac{2 \times 40 \times 60}{100} = 48$ km/h. The figure (40 + 60) ÷ 2 = 50 is wrong: more time is spent at the slower speed, so the average is below the simple mean.
- Equal distances: average speed = 2xy ÷ (x + y)
- Total distance ÷ total time
- Writing (40 + 60) ÷ 2 = 50
- Harmonic mean for equal distances, not the simple mean
| Conversion | How |
|---|---|
| km/h → m/s | Multiply by $\frac{5}{18}$ (72 → 20) |
| m/s → km/h | Multiply by $\frac{18}{5}$ (10 → 36) |
| Relative speed, opposite directions | Sum of the speeds |
| Relative speed, same direction | Difference of the speeds |
The length of a rectangle is 15 m and its breadth is 8 m. What is the length of its diagonal?
Diagonal $= \sqrt{l^{2} + b^{2}} = \sqrt{15^{2} + 8^{2}} = \sqrt{225 + 64} = \sqrt{289} = 17$ m. The figure 23 is length plus breadth, and 120 is the area (in square metres).
- The diagonal comes from Pythagoras' theorem
- (8, 15, 17) is a well-known triple
- Mixing up the diagonal with the perimeter or area
- Remember 8-15-17 and you skip the square root
| Shape | Area | Perimeter | Diagonal |
|---|---|---|---|
| Rectangle | l × b | 2(l + b) | $\sqrt{l^{2} + b^{2}}$ |
| Square | $a^{2}$ | 4a | $a\sqrt{2}$ |
📊 Topic Analysis
One question from each arithmetic topic of the Clerkship 2024 syllabus, with an explanation for every answer.
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