Clerkship Arithmetic Mixed Practice Quiz: WBPSC Clerkship 2024
A little practice. A step forward.
20 questions on Part-I arithmetic: HCF, LCM, average, ratio, percentage, simple interest, profit and loss, time and distance, area and partnership.
What is the HCF of 36, 48 and 60?
36 = $2^{2} \times 3^{2}$, 48 = $2^{4} \times 3$, 60 = $2^{2} \times 3 \times 5$. The common factor with the lowest powers is $2^{2} \times 3 = 12$. 24 is not a factor of 36.
- HCF = product of common prime factors with the smallest powers
- HCF divides every given number
- Mixing up HCF and LCM
- HCF = Highest Common Factor, a divisor of all
| Number | Prime factors |
|---|---|
| 36 | $2^{2} \times 3^{2}$ |
| 48 | $2^{4} \times 3$ |
| 60 | $2^{2} \times 3 \times 5$ |
What is the LCM of 12, 18 and 30?
12 = $2^{2} \times 3$, 18 = $2 \times 3^{2}$, 30 = $2 \times 3 \times 5$. Taking the highest power of each prime gives $2^{2} \times 3^{2} \times 5 = 180$. 90 is not divisible by 12 and 360 is not the least common multiple.
- LCM = product of each prime with its highest power
- LCM is a multiple of every given number
- Highest power for LCM, lowest power for HCF
- LCM = Least Common Multiple, a multiple of all
Which is the smallest number that leaves remainder 3 when divided by 4, 6 and 10?
The LCM of 4, 6 and 10 is 60. For the same remainder 3 each time, the number is 60 + 3 = 63. 60 leaves remainder 0, and 123 also works but is not the smallest.
- Same remainder r: number = LCM + r
- The remainder must be smaller than every divisor
- Taking the LCM (60) itself as the answer
- First the LCM, then add the remainder
The average of 6 numbers is 20. If a seventh number 34 is added, what is the new average?
The sum of the 6 numbers is 6 × 20 = 120. After adding 34 the sum is 154 for 7 numbers, so the average is 154 ÷ 7 = 22. It is not 27, because only one new number is added.
- Average = sum ÷ number of items
- Sum = average × number of items
- Averaging 20 and 34 directly to get 27
- Find the total first, then add the new number
If ₹720 is divided in the ratio 3 : 5 : 4, what is the largest share?
The ratio total is 3 + 5 + 4 = 12, so one part = 720 ÷ 12 = ₹60. The largest share is 5 parts = 5 × 60 = ₹300. The other shares are ₹180 and ₹240, and 180 + 300 + 240 = 720.
- One part = total ÷ sum of ratio terms
- The shares always add up to the total
- Picking the smallest or middle share by mistake
- Add the shares back to check the total
| Share | Ratio | Amount |
|---|---|---|
| First | 3 | ₹180 |
| Second | 5 | ₹300 |
| Third | 4 | ₹240 |
If A : B = 2 : 3 and B : C = 4 : 5, what is A : C?
Make B equal: multiply the first ratio by 4 and the second by 3. A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15 and A : C = 8 : 15. Writing 2 : 5 ignores that B has different values (3 and 4).
- The common term (B) must be made equal
- A : B : C = 8 : 12 : 15
- Writing 2 : 5 without matching the two values of B
- Cross-multiply to match the middle term
45 is what percent of 360?
Percent = (45 ÷ 360) × 100 = $\frac{1}{8} \times 100 = 12.5$%. 10% of 360 would be 36 and 15% would be 54, neither is 45.
- Percent = (part ÷ whole) × 100
- $\frac{1}{8}$ = 12.5%
- Dividing the whole by the part (360 ÷ 45)
- $\frac{1}{8}$ = 12.5%, $\frac{1}{4}$ = 25%, $\frac{1}{2}$ = 50%
A price rises by 20% and then the increased price falls by 20%. What is the net change compared with the original?
Take the original price as 100. After a 20% rise it is 120. A 20% fall of 120 is 24, giving 96. So the net change from 100 to 96 is a 4% decrease. The fall applies to the increased price, so the net change is not zero.
- The rise and the fall use different bases
- Net change for +x% then −x% is $-\frac{x^{2}}{100}$%
- Assuming 20% − 20% = 0
- Assume the price is 100 and go step by step
| Step | Price |
|---|---|
| Start | 100 |
| After 20% rise | 120 |
| After 20% fall | 96 |
What is the simple interest on ₹5,000 for 3 years at 8% per annum?
Simple interest = $\frac{P \times R \times T}{100} = \frac{5000 \times 8 \times 3}{100}$ = ₹1,200. ₹400 is only one year's interest. ₹1,000 and ₹1,500 do not fit the formula.
- SI = $\frac{P \times R \times T}{100}$
- Amount = principal + interest = ₹6,200
- Giving one year's interest as the answer
- P R T over 100
At simple interest a sum doubles in 10 years. What is the annual rate?
When the sum doubles, the interest equals the principal. So $P = \frac{P \times R \times 10}{100}$, giving R = 10%. At 20% it would double in 5 years and at 5% it would take 20 years.
- For doubling, interest = principal
- R = $\frac{100}{T}$
- Mixing up interest and amount
- Rate × time = 100 means the sum doubles
| Years | Interest (R = 10%) | Total |
|---|---|---|
| 0 | 0 | P |
| 5 | 0.5P | 1.5P |
| 10 | P | 2P |
An article bought for ₹800 is sold for ₹920. What is the profit percent?
Profit = 920 − 800 = ₹120. Profit percent = (120 ÷ 800) × 100 = 15%. The rate is always taken on the cost price, not on the selling price.
- Profit% = (profit ÷ cost price) × 100
- SP = CP + profit
- Dividing by the selling price instead of the cost price
- Profit and loss percent are always on the cost price
| Cost price | Profit | Selling price | Profit% |
|---|---|---|---|
| ₹800 | ₹120 | ₹920 | 15% |
An article is sold for ₹600 at a 20% profit. What is its cost price?
120% of the cost price = 600, so cost price = 600 × $\frac{100}{120}$ = ₹500. ₹480 comes from subtracting 20% of 600, which is wrong because the 20% is on the cost price, not the selling price.
- SP = CP × (100 + profit%) ÷ 100
- Check: 20% of 500 = 100, and 500 + 100 = 600
- Subtracting 20% of 600 to get ₹480
- Multiply SP by 100, divide by (100 + profit%)
What is a speed of 72 km/h in metres per second?
To convert km/h to m/s multiply by $\frac{5}{18}$: 72 × $\frac{5}{18}$ = 20 m/s. The reverse conversion multiplies by $\frac{18}{5}$. 18 m/s would be 64.8 km/h.
- km/h to m/s: × $\frac{5}{18}$
- m/s to km/h: × $\frac{18}{5}$
- Multiplying by $\frac{18}{5}$ and getting 129.6
- Multiply by five-eighteenths to get metres per second
| km/h | m/s |
|---|---|
| 18 | 5 |
| 36 | 10 |
| 54 | 15 |
| 72 | 20 |
How long does a 150 m train running at 54 km/h take to cross a pole?
Speed = 54 × $\frac{5}{18}$ = 15 m/s. To cross a pole the train covers its own length, so time = 150 ÷ 15 = 10 seconds. Not converting the speed to m/s gives a wrong number.
- Distance to cross a pole = length of the train
- Time = distance ÷ speed
- Dividing 150 by 54 without matching units
- Convert the speed to m/s first
A person travels one way at 60 km/h and returns along the same route at 40 km/h. What is the average speed for the whole trip?
For equal distances the average speed is $\frac{2xy}{x + y} = \frac{2 \times 60 \times 40}{100}$ = 48 km/h. 50 is the plain mean of the speeds, which does not apply because more time is spent at the slower speed.
- Equal distances: average speed = $\frac{2xy}{x + y}$
- Average speed = total distance ÷ total time
- Assuming (60 + 40) ÷ 2 = 50
- Round-trip average speed is always below the plain mean
| Leg | Distance (say 120 km) | Time |
|---|---|---|
| Going (60 km/h) | 120 km | 2 hours |
| Return (40 km/h) | 120 km | 3 hours |
| Total | 240 km | 5 hours |
Average speed = 240 ÷ 5 = 48 km/h.
A rectangle has a perimeter of 36 m and a length of 10 m. What is its area?
Perimeter = 2 × (length + breadth) = 36, so length + breadth = 18 and breadth = 18 − 10 = 8 m. Area = 10 × 8 = 80 square metres. 100 would be the area of a square of side 10 m.
- Perimeter of a rectangle = 2 × (length + breadth)
- Area = length × breadth
- Using 36 instead of half the perimeter (18) to find the breadth
- Half the perimeter = length + breadth
What is the area of a square whose perimeter is 48 m?
Perimeter of a square = 4 × side, so side = 48 ÷ 4 = 12 m. Area = $12^{2}$ = 144 square metres. 192 is 48 × 4 and 96 is 48 × 2, neither fits.
- Perimeter of a square = 4 × side
- Area of a square = side × side
- Treating 48 as the area and taking its square root
- Perimeter ÷ 4 = side, side squared = area
| Side | Perimeter | Area |
|---|---|---|
| 10 m | 40 m | 100 sq m |
| 12 m | 48 m | 144 sq m |
| 13 m | 52 m | 169 sq m |
A and B start a business with ₹6,000 and ₹9,000 for the same period. If the total profit is ₹7,500, what is A's share?
For equal periods the profit is shared in the ratio of capital: 6000 : 9000 = 2 : 3. A's share = 7500 × $\frac{2}{5}$ = ₹3,000, and B gets ₹4,500. ₹4,500 is B's share.
- Equal time: profit ratio = capital ratio
- Unequal time: ratio = capital × time
- Giving B's share (₹4,500) as A's
- Reduce the capitals to the simplest ratio (2 : 3)
What is the value of $\frac{3}{4} + \frac{5}{6}$?
The LCM of the denominators 4 and 6 is 12. $\frac{3}{4} = \frac{9}{12}$ and $\frac{5}{6} = \frac{10}{12}$, so the sum is $\frac{19}{12} = 1\frac{7}{12}$. $\frac{8}{10}$ comes from adding numerators and denominators separately, which is wrong.
- Find the LCM of the denominators first
- An improper fraction can be written as a mixed number
- Adding numerators and denominators separately
- Make the denominators equal before adding
What is the value of $15 + 5 \times 4 - 6 \div 2$?
By BODMAS, multiplication and division come first: 5 × 4 = 20 and 6 ÷ 2 = 3. Then add and subtract: 15 + 20 − 3 = 32. Working strictly left to right gives the wrong value 77.
- BODMAS: Brackets, Of, Division, Multiplication, Addition, Subtraction
- Division and multiplication share a level, as do addition and subtraction
- Calculating strictly from left to right
- Multiply and divide first, add and subtract after
📊 Topic Analysis
20 questions on Part-I arithmetic: HCF, LCM, average, ratio, percentage, simple interest, profit and loss, time and distance, area and partnership.
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