Number System & Arithmetic MCQs With Answers
Every quantitative section starts with numbers, and quick, error-free calculation is what separates a pass from a merit. These number system and arithmetic MCQs cover BODMAS and order of operations, fractions and decimals, squares, cubes, square roots and indices, prime and composite numbers, factors, HCF and LCM, divisibility and remainders, unit digits of large powers, factorials and unit conversions. They are built for the basic maths portion of PPSC, FPSC, NTS, OTS, ETEA, KPPSC and SPSC tests, as well as entry tests and school-level revision.
Take a scored quiz instead →Solve: 2 + 2 * 2?
• According to BODMAS/PEMDAS rule, multiplication is performed before addition.
• 2 + (2 * 2) = 2 + 4 = 6.
What is the square root of 144?
• 12 * 12 = 144. Therefore, sqrt(144) = 12.
What is 1/2 + 1/4?
• Convert to common denominator: 2/4 + 1/4 = 3/4.
A shopkeeper buys 10 pens at Rs 45 each and sells all of them at Rs 52 each. What is his total profit?
Profit on one pen = Rs 52 − Rs 45 = Rs 7.
Total profit on 10 pens = 10 × Rs 7 = Rs 70.
Arrange 3/8, 0.35, 2/5 and 5/13 in ascending order.
Convert each to a decimal: 3/8 = 0.375, 2/5 = 0.4 and 5/13 ≈ 0.385.
So the ascending order is 0.35 < 0.375 < 0.385 < 0.4, that is 0.35, 3/8, 5/13, 2/5.
Solve: 10 – (2 * 3)?
• Multiplication first: 2 * 3 = 6.
• 10 - 6 = 4.
What is the value of (2⁵ × 2³) ÷ 2⁶?
When multiplying powers with the same base, add the indices; when dividing, subtract them.
(2⁵ × 2³) ÷ 2⁶ = 2⁵⁺³⁻⁶ = 2² = 4.
What is the cube root of 2,744?
2,744 = 2 × 2 × 2 × 7 × 7 × 7 = (2 × 7)³ = 14³.
So ∛2,744 = 14.
How many seconds are there in 2 hours 15 minutes?
2 hours 15 minutes = 2 × 60 + 15 = 135 minutes.
135 minutes × 60 = 8,100 seconds.
What is the value of |−7| − |3 − 9| + |−2|?
The absolute value of a number is its distance from zero, so it is never negative: |−7| = 7, |3 − 9| = |−6| = 6 and |−2| = 2.
So the expression = 7 − 6 + 2 = 3.
A train leaves at 22:45 and the journey takes 5 hours 40 minutes. At what time does it arrive?
22:45 + 5 hours = 03:45 the next day.
03:45 + 40 minutes = 04:25, so the train arrives at 04:25 (4:25 a.m.) the next day.
Which number is a prime number?
• A prime number has only two factors: 1 and itself. 7 is prime.
What is 0.1 * 0.1?
• 0.1 * 0.1 = 0.01.
Simplify 20/4 + 2?
• 20/4 = 5. 5 + 2 = 7.
Solve: 8 – 4 + 2?
• Follow left-to-right: 8 - 4 = 4. 4 + 2 = 6.
What is 2^4?
• 2 * 2 * 2 * 2 = 16.
Which fraction is commonly used as an approximation of π (pi)?
22/7 = 3.142857…, which is close to π = 3.14159…, so it is the standard approximation used in calculations.
A more accurate fraction is 355/113 = 3.1415929…
How many items are there in a gross?
A gross is a dozen dozens: 12 × 12 = 144.
A score is 20, and a great gross is 12 gross = 1,728.
Solve: 100 / 10 * 2?
• Left to right: 100 / 10 = 10. 10 * 2 = 20.
Simplify: 15 × 4 − 36 ÷ 6 + 2³
By BODMAS, work out the power first, then division and multiplication, then addition and subtraction.
2³ = 8, 15 × 4 = 60 and 36 ÷ 6 = 6, so the value = 60 − 6 + 8 = 62.
What is the value of (7 × 0) + (0 ÷ 7) + 7⁰?
7 × 0 = 0 and 0 ÷ 7 = 0. Any non-zero number raised to the power 0 equals 1, so 7⁰ = 1.
So the value = 0 + 0 + 1 = 1. (Division by zero, such as 7 ÷ 0, is what is not defined.)
How many minutes are there in one week?
One day = 24 × 60 = 1,440 minutes.
One week = 7 × 1,440 = 10,080 minutes.
Simplify using BODMAS: 20 + 30 ÷ 5 × 2 − 10
Division and multiplication are done from left to right before addition and subtraction: 30 ÷ 5 = 6, then 6 × 2 = 12.
So the value = 20 + 12 − 10 = 22.
What is the value of 99²?
Use (a − b)² = a² − 2ab + b² with a = 100 and b = 1.
99² = 10,000 − 200 + 1 = 9,801.
Which of the following fractions is the largest?
Convert to decimals: 7/10 = 0.700, 9/13 ≈ 0.692, 5/7 ≈ 0.714 and 11/16 = 0.6875.
So 5/7 is the largest.
Solve: 5 * (2 + 3)?
• Parentheses first: 2 + 3 = 5. Then 5 * 5 = 25.
What is the square root of 169?
13 * 13 = 169.
Solve: 5 + 3 * 2.
5 + (3 * 2) = 5 + 6 = 11.
A machine fills one bottle every 45 seconds. How many bottles does it fill in 3 hours?
3 hours = 3 × 3,600 = 10,800 seconds.
Number of bottles = 10,800 ÷ 45 = 240.
Smallest even composite number?
4 is the smallest composite number.
What is 0.5 * 0.2?
0.1.
Simplify: 100 − [50 − {25 − (10 − 4)}]
Remove the brackets from the innermost outwards: (10 − 4) = 6, then {25 − 6} = 19, then [50 − 19] = 31.
So the value = 100 − 31 = 69.
Smallest prime number?
2.
How many prime numbers between 1 and 10?
2, 3, 5, 7.
Solve: 10 + 10 * 2.
10 + (10 * 2) = 10 + 20 = 30.
A runner completes 12 laps of a 400 m track. How many kilometres has he run?
Total distance = 12 × 400 m = 4,800 m.
Since 1 km = 1,000 m, 4,800 m = 4.8 km.
What is the value of 0.1 + 0.01 + 0.001 + 0.0001?
Write the numbers with the same number of decimal places: 0.1000 + 0.0100 + 0.0010 + 0.0001.
The sum is 0.1111.
Sum of first 4 even numbers?
2+4+6+8=20.
Convert 4.35 hours into hours and minutes.
The decimal part is a fraction of an hour, not minutes: 0.35 hour = 0.35 × 60 = 21 minutes.
So 4.35 hours = 4 hours 21 minutes.
Simplify: 5 + 10 × 3 − 18 ÷ 3²
Powers first: 3² = 9. Then 10 × 3 = 30 and 18 ÷ 9 = 2.
So the value = 5 + 30 − 2 = 33.
Smallest composite number?
4.
What is the smallest number by which 72 must be multiplied to make it a perfect square?
72 = 2³ × 3². For a perfect square, every prime factor must have an even power.
The factor 2 has power 3, so multiply by 2: 72 × 2 = 144 = 12².
Express 0.375 as a fraction in its lowest terms.
0.375 = 375/1,000.
Dividing numerator and denominator by their HCF, 125, gives 3/8.
Sum of first 5 natural numbers?
1+2+3+4+5=15.
The value of 3^0 + 4^0 + 5^0 is:
1 + 1 + 1 = 3.
Find the LCM of 4, 8, and 12.
Multiples: 24 is the smallest common.
Which of these is a prime number?
23 is only divisible by 1 and itself.
What is the value of √0.0064?
0.0064 = 64/10,000.
√(64/10,000) = 8/100 = 0.08. (Check: 0.08 × 0.08 = 0.0064.)
What is 10 * 10 / 2 – 5?
100 / 2 - 5 = 50 - 5 = 45.
How many minutes in 1.5 hours?
1.5 * 60 = 90.
If 2^(x + 1) = 64, what is the value of x?
64 = 2⁶, so 2^(x + 1) = 2⁶.
Equating the indices: x + 1 = 6, so x = 5.
Simplify: 2/3 + 3/4 − 5/6
The LCM of 3, 4 and 6 is 12, so write each fraction over 12: 8/12 + 9/12 − 10/12.
The result is 7/12.
What is 5!?
5 * 4 * 3 * 2 * 1 = 120.
Solve 5 + 5 * 5.
5 + 25 = 30.
Which of the following fractions lies between 1/3 and 1/2?
1/3 ≈ 0.333 and 1/2 = 0.5. Converting the options: 1/4 = 0.25, 2/7 ≈ 0.286, 3/7 ≈ 0.429 and 4/7 ≈ 0.571.
Only 3/7 lies between 0.333 and 0.5.
Sum of first 3 even numbers:
2 + 4 + 6 = 12.
A journey takes 2 hours 48 minutes. What fraction of a day is this?
2 hours 48 minutes = 168 minutes, and one day = 1,440 minutes.
168/1,440 = 7/60.
Simplify: 10 + 20 ÷ 4 × 2 − 5
Division and multiplication from left to right first: 20 ÷ 4 = 5, then 5 × 2 = 10.
So the value = 10 + 10 − 5 = 15.
The sum of the squares of two consecutive natural numbers is 85. What are the numbers?
Check the options: 5² + 6² = 61, 6² + 7² = 36 + 49 = 85, 7² + 8² = 113.
So the numbers are 6 and 7.
What is the value of 0.5 ÷ 0.025?
Multiply both numbers by 1,000 to remove the decimals: 500 ÷ 25.
500 ÷ 25 = 20.
What is the square of 11?
11*11=121.
How many minutes in 2.5 hours?
2.5 * 60 = 150.
Smallest even prime number?
2.
Simplify: 10 + 2 * 3 – 4.
10 + 6 - 4 = 12.
What is 1/5 + 2/5?
3/5 = 0.6.
What is 2^4 * 2^2?
2^(4+2) = 2^6.
A ribbon 3.6 m long is cut into pieces of 45 cm each. How many pieces are obtained?
Convert to the same unit: 3.6 m = 360 cm.
Number of pieces = 360 ÷ 45 = 8.
Sum of first 4 prime numbers:
2+3+5+7=17.
Find value of 5 + 5 * 2.
5+10=15.
Which is 0.2 as a fraction?
0.2 = 2/10 = 1/5.
The difference between two numbers is 10 and their sum is 50. What is the smaller number?
x+y=50, x-y=10 => 2x=60, x=30, y=20.
What is the value of 2^5 * 2^3 / 2^4?
2^(5+3-4) = 2^4 = 16.
Which of these is a composite number?
9 is divisible by 1, 3, 9.
What is the square root of 0.09?
0.3 * 0.3 = 0.09.
Simplify: 2 + 2 * 2 – 2.
2 + 4 - 2 = 4.
What is 1/4 + 1/8?
2/8 + 1/8 = 3/8 = 0.375.
How many prime numbers between 10 and 20?
11, 13, 17, 19.
What is 4^2 * 4^0?
16 * 1 = 16.
What is the cube of 5?
5*5*5=125.
What is the value of 0.3 × 0.3 × 0.3?
3 × 3 × 3 = 27, and the three factors have one decimal place each, so the answer has three decimal places.
0.3 × 0.3 × 0.3 = 0.027.
A yard is approximately:
Fact.
Which of the following is the smallest?
Compare the digits place by place: 0.07, 0.0701 and 0.0699 all have about 7 hundredths, but 0.007 has 0 hundredths.
So 0.007 is the smallest.
Sum of first 3 odd numbers:
1+3+5=9.
Find value of 20 + 2 * 5.
20+10=30.
What is the value of 25² − 15²?
Use a² − b² = (a + b)(a − b).
25² − 15² = (25 + 15)(25 − 15) = 40 × 10 = 400.
Which is 0.75 as a fraction?
0.75 = 75/100 = 3/4.
The sum of three consecutive odd numbers is 57. What is the middle number?
x-2 + x + x+2 = 57 => 3x=57 => x=19.
Find the value of 5! – 4!.
120 - 24 = 96.
Which of these is an irrational number?
sqrt(3) cannot be expressed as a fraction.
Square root of 1024 is:
32*32=1024.
Simplify: 5 + 5 * (5 – 5).
5 + 5 * 0 = 5.
What is 1/2 + 1/3 + 1/6?
3/6 + 2/6 + 1/6 = 6/6 = 1.
How many prime numbers between 50 and 60?
53, 59.
What is 3^3 * 3^1?
3^(3+1) = 3^4.
What is the cube of 4?
4*4*4=64.
What is 0.25 * 40?
10.
A car travels at 72 km/h. What is its speed in metres per second?
To convert km/h to m/s, multiply by 1,000/3,600 = 5/18.
72 × 5/18 = 20 m/s.
What is 12 squared?
12*12=144.
Find the value of (2^3)^2.
2^(3*2) = 2^6 = 64.
What is the value of 0! + 2^0?
1 + 1 = 2.
Which is a prime number?
41 is not divisible by any number but 1 and itself.
What is 0.1% of 2,500?
0.1% = 0.1/100 = 0.001.
0.001 × 2,500 = 2.5.
If 1 January of a non-leap year is a Monday, what day of the week is 1 March of that year?
From 1 January to 1 March there are 31 (January) + 28 (February) = 59 days.
59 ÷ 7 leaves a remainder of 3 (odd days), so 1 March falls 3 days after Monday, on Thursday.
What is 50^2?
50*50=2500.
What is the sum of the first 50 natural numbers?
The sum of the first n natural numbers = n(n + 1)/2.
For n = 50: 50 × 51 ÷ 2 = 1,275.
Arrange 0.19, 0.2, 0.091 and 0.109 in descending order.
Write them with three decimal places: 0.190, 0.200, 0.091 and 0.109.
From largest to smallest: 0.200, 0.190, 0.109, 0.091.
What is the value of 7⁰ + 0⁷ + 1⁷?
7⁰ = 1 (any non-zero number to the power 0 is 1), 0⁷ = 0 and 1⁷ = 1.
So the value = 1 + 0 + 1 = 2.
Simplify: 6 × 2 + 4 × (8 − 3)² ÷ 10
Brackets first: (8 − 3) = 5, and 5² = 25. Then 4 × 25 ÷ 10 = 10 and 6 × 2 = 12.
So the value = 12 + 10 = 22.
What is the value of √(3² + 4²) + √(5² + 12²)?
√(9 + 16) = √25 = 5 and √(25 + 144) = √169 = 13.
So the value = 5 + 13 = 18.
Find the sum of the first 20 odd numbers.
Sum = n^2 = 20^2 = 400.
Which is greater: 2/3 or 3/4?
0.66 < 0.75.
What is the square root of 1.44?
1.2 * 1.2 = 1.44.
The sum of the first 5 prime numbers is:
2+3+5+7+11 = 28.
Find the value of 0.2 + 0.22 + 0.222.
Arithmetic addition.
What is the value of 999 × 999?
Write 999 as (1,000 − 1) and use (a − b)² = a² − 2ab + b².
999² = 1,000,000 − 2,000 + 1 = 998,001.
What is 50 divided by 0.5?
50 / 0.5 = 100.
Which of the following fractions is the smallest?
Convert to decimals: 3/5 = 0.600, 4/7 ≈ 0.571, 5/9 ≈ 0.556 and 7/12 ≈ 0.583.
So 5/9 is the smallest.
Value of 4 + 4 * 4.
4+16=20.
What is the square root of 1,296?
1,296 = 2⁴ × 3⁴ = (2² × 3²)² = 36².
So √1,296 = 36.
What is 15 * 15?
225.
Find the value of sqrt(100) + sqrt(64).
10 + 8 = 18.
What is 10^0?
Anything to power 0 is 1.
How many hours are there from 9:00 a.m. on Monday to 3:00 p.m. on Thursday of the same week?
From 9:00 a.m. Monday to 9:00 a.m. Thursday is 3 days = 72 hours.
From 9:00 a.m. to 3:00 p.m. Thursday is 6 more hours, so the total is 78 hours.
Sum of the first 4 odd numbers:
1+3+5+7 = 16.
Simplify: 2 * 3 + 4 / 2.
6 + 2 = 8.
What is the value of 1,000 ÷ 10 ÷ 0.5 × 2?
Division and multiplication are worked from left to right: 1,000 ÷ 10 = 100, then 100 ÷ 0.5 = 200.
Then 200 × 2 = 400.
What is the value of (−5) + 10 − (−8) − 12?
Subtracting a negative number is the same as adding it: −(−8) = +8.
So −5 + 10 + 8 − 12 = 1.
Which of the following is the largest?
Write each with four decimal places: 0.5050, 0.5500, 0.5505 and 0.5550.
The largest is 0.5550, that is 0.555.
How many seconds are there in a leap year?
One day has 24 × 60 × 60 = 86,400 seconds, and a leap year has 366 days.
366 × 86,400 = 31,622,400 seconds. (A non-leap year has 31,536,000.)
Simplify: 10 + 2 × 5 − 8 ÷ 4 + 3²
Powers first: 3² = 9. Then 2 × 5 = 10 and 8 ÷ 4 = 2.
So the value = 10 + 10 − 2 + 9 = 27.
Express 7/40 as a decimal.
Make the denominator a power of 10: 7/40 = (7 × 25)/(40 × 25) = 175/1,000.
175/1,000 = 0.175.
Which of these is a rational number?
0.5 = 1/2.
What is the square root of 625?
25*25=625.
Simplify: 10 + 2 * (5 – 3).
10 + 2 * 2 = 14.
Simplify: 3/4 + 5/6 − 7/12
The LCM of 4, 6 and 12 is 12: 9/12 + 10/12 − 7/12 = 12/12.
So the value is 1.
How many prime numbers between 20 and 30?
23, 29.
What is 12.5% of 640?
12.5% = 12.5/100 = 1/8.
1/8 of 640 = 80.
A US gallon is approximately equal to:
One US liquid gallon is about 3.785 litres. The imperial (UK) gallon is larger, about 4.546 litres.
Which is the smallest: 1/5, 1/4, 1/3?
0.2 < 0.25 < 0.33.
How many seconds are there in 1 day?
24 * 60 * 60 = 86400.
Find the value of sqrt(169) – sqrt(144).
13 - 12 = 1.
The sum of the first 10 natural numbers is:
n(n+1)/2 = 10*11/2 = 55.
The value of 10! / 8! is:
10*9 = 90.
What is 100 degrees Fahrenheit in Celsius (approx)?
(100-32)*5/9 = 68*5/9 = 37.7.
The smallest two-digit prime number is:
11.
If a man earns 3000 in 15 days, what does he earn in a day?
3000/15 = 200.
What is 10^3 + 10^2?
1000 + 100 = 1100.
Simplify 1/2 + 1/3.
3/6 + 2/6 = 5/6.
What is the value of (3⁰ + 5⁰) × (2⁻¹ + 2⁻²)?
3⁰ = 5⁰ = 1, so the first bracket is 2. A negative index means a reciprocal: 2⁻¹ = 1/2 and 2⁻² = 1/4, so the second bracket is 3/4.
So the value = 2 × 3/4 = 1.5.
What is 0.1 divided by 0.01?
10.
What is the value of 0.01 ÷ 0.0001?
Multiply both numbers by 10,000 to remove the decimals: 100 ÷ 1.
So 0.01 ÷ 0.0001 = 100.
On which day of the week did 14 August 1947 fall?
Pakistan's Independence Day, 14 August 1947, fell on a Thursday (it was also 27 Ramadan 1366 AH).
Such questions are solved by counting odd days: 1946 years have 2 odd days (1600 years: 0, 300 years: 1, 46 years: 11 leap + 35 ordinary = 57 days, 1 odd day), and 1 January to 14 August 1947 adds 226 days = 2 odd days, giving 4 odd days = Thursday.
A sum of Rs 5,000 is divided among A, B and C in the ratio 2 : 3 : 5. What is C’s share?
Total parts = 2 + 3 + 5 = 10, so one part = Rs 5,000 ÷ 10 = Rs 500.
C's share = 5 × Rs 500 = Rs 2,500.
Each of 14 boxes holds 3 dozen eggs. If 2 boxes are broken and thrown away, how many eggs remain?
Each box holds 3 × 12 = 36 eggs, and 14 − 2 = 12 boxes remain.
Eggs remaining = 12 × 36 = 432.
What is the value of (0.05)^2?
0.05 * 0.05 = 0.0025.
Which of the following is a composite number?
15 is divisible by 3 and 5.
The value of 0! + 1! + 2! is:
1 + 1 + 2 = 4.
How many minutes are in 3.5 hours?
3.5 * 60 = 210.
Find the value of sqrt(121) – sqrt(81).
11 - 9 = 2.
The decimal equivalent of 3/8 is:
3 / 8 = 0.375.
Simplify: 15 + 5 * 2 – 4.
15 + 10 - 4 = 21.
How many prime numbers are there up to 20?
2, 3, 5, 7, 11, 13, 17, 19.
From a 1,000-litre tank, 37.5% of the water is used on Monday and 40% of the remainder on Tuesday. How many litres are left?
After Monday: 1,000 × (1 − 0.375) = 625 litres remain.
After Tuesday: 625 × (1 − 0.40) = 375 litres remain.
What is the value of 10 × 10 ÷ 0.1 × 0.01?
Work from left to right: 10 × 10 = 100, then 100 ÷ 0.1 = 1,000.
Then 1,000 × 0.01 = 10.
What is the value of 0.2 × 0.02 × 0.002?
2 × 2 × 2 = 8, and the factors have 1 + 2 + 3 = 6 decimal places in total.
So the product is 0.000008.
Find the value of (2^3 * 2^4) / 2^5.
2^(3+4-5) = 2^2 = 4.
What is the square root of 0.0001?
0.01*0.01 = 0.0001.
What is the value of 0.5 + 0.05 + 0.005?
Arithmetic addition.
The number of prime numbers between 1 and 50 is:
Count is 15.
Through how many degrees does the second hand of a clock turn in 5 minutes?
The second hand makes one full turn (360°) every minute.
In 5 minutes it turns 5 × 360° = 1,800°.
The HCF of 12, 15, and 18 is:
3 is the common factor.
Simplify: 5 * (2 + 3) – 10.
5*5-10 = 15.
Find the value of sqrt(16) + sqrt(9).
4+3=7.
The smallest whole number is:
Definition.
What is the decimal form of 5/16?
16 = 2⁴, so multiply numerator and denominator by 5⁴ = 625: 5/16 = 3,125/10,000.
So 5/16 = 0.3125.
Simplify 12/16.
Divide by 4.
If x² = 625 and x is positive, what is the value of √x?
x² = 625 gives x = 25 (taking the positive root).
So √x = √25 = 5.
Solve: 8 – 4 / 2.
8 - 2 = 6.
How many months have 31 days?
Jan, Mar, May, Jul, Aug, Oct, Dec.
A number when divided by 6 leaves a remainder of 3. What will be the remainder when the square of this number is divided by 6?
Let the number be 9 (9/6 rem 3). Square is 81. 81/6 = 13*6 + 3. Remainder is 3.
A man buys 5 chairs and 3 tables for Rs. 3500. If a table costs Rs. 500, what is the cost of a chair?
5C + 3(500) = 3500 => 5C + 1500 = 3500 => 5C = 2000 => C = 400.
What is the cube root of 27000?
30*30*30 = 27000.
The HCF of 15 and 20 is:
5 divides both.
By how much does 1/3 exceed 1/4?
Using the LCM 12: 1/3 = 4/12 and 1/4 = 3/12.
Difference = 4/12 − 3/12 = 1/12.
Square of 0.2 is:
0.2*0.2=0.04.
For how many integers x is |x − 3| ≤ 2?
|x − 3| ≤ 2 means −2 ≤ x − 3 ≤ 2, so 1 ≤ x ≤ 5.
The integers are 1, 2, 3, 4 and 5, which is 5 values.
Pens cost Rs 180 per dozen. What is the greatest number of pens that can be bought with Rs 1,000, and how much money is left?
Price of one pen = Rs 180 ÷ 12 = Rs 15.
Rs 1,000 ÷ 15 = 66 with remainder 10, so 66 pens can be bought (cost Rs 990) and Rs 10 is left.
Which of the following fractions is equivalent to 18/24?
The HCF of 18 and 24 is 6.
18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4.
If 5 workers earn Rs 7,500 in 3 days, how much will 8 workers earn in 5 days at the same daily rate?
Daily wage of one worker = Rs 7,500 ÷ (5 × 3) = Rs 500.
8 workers for 5 days = 40 worker-days × Rs 500 = Rs 20,000.
The sum of three consecutive integers is 51. Find the largest integer.
Let integers be x-1, x, x+1. 3x = 51 => x = 17. Largest = 17+1=18.
Find the HCF of 24, 36, and 48.
All are divisible by 12.
How many seconds are there in 2 hours?
2 * 60 * 60 = 7200.
What is the unit digit of 2²⁵?
The unit digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6.
25 ÷ 4 leaves a remainder of 1, so 2²⁵ has the same unit digit as 2¹, which is 2.
What is the sum of the first 10 even numbers?
n(n+1) = 10*11 = 110.
The value of 0! is:
Mathematical definition.
Which of the following numbers is a prime number?
91 = 7 × 13, 119 = 7 × 17 and 133 = 7 × 19, so all three are composite.
127 is not divisible by any prime up to √127 ≈ 11.3 (that is 2, 3, 5, 7 or 11), so 127 is prime.
The difference between a two-digit number and the number formed by reversing its digits is 36. What is the difference between its two digits?
If the digits are a and b, the numbers are 10a + b and 10b + a, and their difference is 9(a − b).
9(a − b) = 36, so a − b = 4.
What is the square of 15?
15*15=225.
A boy has 5 apples, eats 2. Fraction left?
3/5 left.
Reduce 391/667 to its lowest terms.
Find the HCF: 391 = 17 × 23 and 667 = 23 × 29, so the HCF is 23.
Dividing both by 23 gives 17/29.
Simplify: 10 × 5 + 2 × (12 − 7)² − 40 ÷ 8
Brackets and powers first: (12 − 7)² = 25, so 2 × 25 = 50. Then 10 × 5 = 50 and 40 ÷ 8 = 5.
So the value = 50 + 50 − 5 = 95.
The HCF of two numbers is 11 and their LCM is 7700. If one number is 275, find the other.
HCF * LCM = Product of two numbers. 11 * 7700 = 275 * x => x = 84700 / 275 = 308.
Find the LCM of 8, 12, and 16.
Multiples of 16: 16 (no), 32 (no), 48 (yes).
What is the sum of all odd numbers from 1 to 49?
There are 25 odd numbers from 1 to 49, and the sum of the first n odd numbers is n².
So the sum = 25² = 625.
What is the value of 12² + 13² + 14²?
12² = 144, 13² = 169 and 14² = 196.
144 + 169 + 196 = 509.
0.1 * 0.2?
0.02.
The sum of the digits of a 2-digit number is 9. If the digits are interchanged, the new number is 45 less than the original. Find the number.
Original number 10x + y. x+y=9. New number 10y + x. (10x+y) - (10y+x) = 45 => 9(x-y)=45 => x-y=5. x+y=9, x-y=5 => x=7, y=2. Number is 72.
Find the value of (2^3 * 2^4 * 2^5) / 2^10.
Numerator = 2^(3+4+5) = 2^12. Division = 2^(12-10) = 2^2 = 4.
What is the HCF of 16, 24, and 32?
Divisors of 16: 1, 2, 4, 8, 16. 24: 1, 2, 3, 4, 6, 8, 12, 24. 32: 1, 2, 4, 8, 16, 32. Max is 8.
Simplify: sqrt(0.0025).
sqrt(25/10000) = 5/100 = 0.05.
0.3 * 0.4?
3*4=12, 2 decimal places = 0.12.
The LCM of 12, 15, and 20 is:
Multiples of 20: 20, 40, 60 (divisible by 12 and 15). LCM is 60.
Unit digit of 2^50?
Cyclicity of 2 is 4 (2, 4, 8, 6). 50 mod 4 = 2. 2^2 = 4.
Value of sqrt(5 + sqrt(11 + sqrt(19 + sqrt(29 + sqrt(49)))))?
sqrt(49)=7, sqrt(29+7)=6, sqrt(19+6)=5, sqrt(11+5)=4, sqrt(5+4)=3.
What is the sum of all multiples of 5 from 5 to 100?
The multiples are 5, 10, …, 100, that is 5 × (1 + 2 + … + 20).
1 + 2 + … + 20 = 20 × 21 ÷ 2 = 210, so the sum = 5 × 210 = 1,050.
What is the value of 205²?
Use (a + b)² = a² + 2ab + b² with a = 200 and b = 5.
205² = 40,000 + 2,000 + 25 = 42,025.
What is the unit digit of 3^4?
3 * 3 * 3 * 3 = 81. Unit digit is 1.
Find the value of sqrt(144) + sqrt(64).
12 + 8 = 20.
The HCF of two numbers is 12 and their LCM is 72. If one number is 24, find the other.
- For any two numbers, first number × second number = HCF × LCM.
- Put in the values: 24 × x = 12 × 72 = 864, so x = 864 ÷ 24 = 36.
- Verify the pair: the HCF of 24 and 36 is 12 and their LCM is 72, so 36 satisfies both conditions.
- Option A 24 is wrong because the HCF of 24 and 24 would be 24, not 12.
- This rule holds only for two numbers. For three or more numbers the product of the HCF and LCM is not equal to the product of the numbers.
- HCF and LCM questions appear often in NTS and PPSC maths sections, and this single formula solves most of them.
What is the unit digit of 7^105?
105 mod 4 = 1. 7^1 = 7.
Find the value of sqrt(16) * sqrt(64).
4 * 8 = 32.
Square root of 225?
15^2 = 225.
Simplify: 0.2 * 0.3.
2 * 3 = 6, decimal moves 2 places = 0.06.
Find the unit digit of (7^95 – 3^58).
7^95: 95/4 rem 3, 7^3 ends in 3. 3^58: 58/4 rem 2, 3^2 ends in 9. 3-9 -> borrow -> 13-9 = 4.
Evaluate (1/64)^(-2/3).
(1/64)^(-2/3) = (64)^(2/3) = (4^3)^(2/3) = 4^2 = 16.
Simplify 0.5 * 0.5 + 0.25?
- Multiplication comes before addition, so start with 0.5 × 0.5 = 0.25.
- Now add the second term: 0.25 + 0.25 = 0.5.
- BODMAS places multiplication ahead of addition, which fixes the order of the two steps.
- Adding first would give 0.5 × 0.75 = 0.375, a value that does not appear among the options.
- In fraction form the working is (1/2 × 1/2) + 1/4 = 1/4 + 1/4 = 1/2, which confirms the answer.
The HCF of two numbers is 16 and their product is 3072. What is their LCM?
Product of numbers = HCF * LCM => 3072 = 16 * LCM => LCM = 3072 / 16 = 192.
What is the unit digit in the product (2467)^153 * (341)^72?
Unit digit of 2467^153: cyclicity of 7 is 4. 153 mod 4 = 1 => 7^1 = 7. Unit digit of 341^72 is 1. Product unit digit = 7 * 1 = 7.
Find the value of (256)^(0.16) * (256)^(0.09).
(256)^(0.16 + 0.09) = (256)^(0.25) = (256)^(1/4) = (4^4)^(1/4) = 4.
Find the value of sqrt(30 + sqrt(30 + sqrt(30 + … infinity))).
x = sqrt(30 + x) => x^2 - x - 30 = 0 => (x - 6)(x + 5) = 0 => x = 6.
Simplify: (0.87 * 0.87 * 0.87 + 0.13 * 0.13 * 0.13) / (0.87 * 0.87 – 0.87 * 0.13 + 0.13 * 0.13).
Expression is of form (a^3 + b^3) / (a^2 - ab + b^2) = a + b. Here a = 0.87, b = 0.13. Sum = 0.87 + 0.13 = 1.00.
The product of two numbers is 2025 and their HCF is 15. What is their LCM?
Product of two numbers = HCF * LCM. Therefore, 2025 = 15 * LCM => LCM = 2025 / 15 = 135.
What is the remainder when 2^31 is divided by 5?
2^1 mod 5 = 2, 2^2 mod 5 = 4, 2^3 mod 5 = 3, 2^4 mod 5 = 1. Pattern repeats every 4 powers. Exponent 31 mod 4 = 3. So 2^31 mod 5 = 2^3 mod 5 = 3.
Evaluate: (81)^(3/4) * (27)^(-2/3).
(81)^(3/4) = (3^4)^(3/4) = 3^3 = 27. (27)^(-2/3) = (3^3)^(-2/3) = 3^-2 = 1/9. Product = 27 * (1/9) = 3.
The sum of the squares of three consecutive natural numbers is 110. Find the smallest number.
Let numbers be n-1, n, n+1. (n-1)^2 + n^2 + (n+1)^2 = 110 => 3n^2 + 2 = 110 => 3n^2 = 108 => n^2 = 36 => n = 6. Smallest number is n - 1 = 5.
Find the value of sqrt(20 + sqrt(20 + sqrt(20 + … infinity))).
x = sqrt(20 + x) => x^2 - x - 20 = 0 => (x - 5)(x + 4) = 0 => x = 5.
Find the sum of all two-digit numbers divisible by 5.
Two-digit numbers divisible by 5: 10, 15, 20, ..., 95. First term a = 10, last term l = 95, common difference d = 5. Number of terms n = (95 - 10)/5 + 1 = 18. Sum = (n/2)*(a + l) = 9 * (10 + 95) = 9 * 105 = 945.
Simplify: (0.75 * 0.75 * 0.75 – 0.25 * 0.25 * 0.25) / (0.75 * 0.75 + 0.75 * 0.25 + 0.25 * 0.25).
Expression is (a^3 - b^3) / (a^2 + ab + b^2) = a - b. Here a = 0.75, b = 0.25. Result = 0.75 - 0.25 = 0.50.
Find the LCM of 0.6, 1.2, and 0.18.
Convert to whole numbers with same decimal places: 0.60, 1.20, 0.18 -> 60, 120, 18. LCM of 60, 120, 18 is 360. Divide by 100 to get decimal back: 3.6.
Find the total number of factors of 360.
360 = 2^3 * 3^2 * 5^1. Total factors = (3+1) * (2+1) * (1+1) = 4 * 3 * 2 = 24.
Find the value of (125)^(2/3) * (25)^(-1/2).
(125)^(2/3) = (5^3)^(2/3) = 5^2 = 25. (25)^(-1/2) = (5^2)^(-1/2) = 5^-1 = 1/5. Product = 25 * (1/5) = 5.
Find the value of sqrt(12 + sqrt(12 + sqrt(12 + … infinity))).
x = sqrt(12 + x) => x^2 - x - 12 = 0 => (x - 4)(x + 3) = 0 => x = 4.
Simplify: (0.2 * 0.2 * 0.2 + 0.8 * 0.8 * 0.8) / (0.2 * 0.2 – 0.2 * 0.8 + 0.8 * 0.8).
Expression is of form (a^3 + b^3) / (a^2 - ab + b^2) = a + b. Here a = 0.2, b = 0.8. Sum = 0.2 + 0.8 = 1.0.
Find the value of sqrt(6 + sqrt(6 + sqrt(6 + … infinity))).
Let x = sqrt(6 + x) => x^2 = 6 + x => x^2 - x - 6 = 0 => (x - 3)(x + 2) = 0 => x = 3.
If 15 articles cost Rs. 225, how much will 28 articles cost?
Cost per article = 225 / 15 = Rs. 15. Cost of 28 articles = 28 * 15 = Rs. 420.
Evaluate: 0.008^(2/3).
- Spot the cube first: 0.008 = 0.2 × 0.2 × 0.2, so 0.008 = (0.2)³.
- Now the power is easy: ((0.2)³)^(2/3) = 0.2² = 0.04.
- Take the cube root before squaring. Working in that order keeps the numbers small and avoids errors.
- Option C 0.2 is only the cube root, so it stops one step too early.
- Counting decimal places is a fast check: 0.2 has one place, and squaring it must give two, which matches 0.04.
Find the smallest number by which 3600 must be divided to make it a perfect cube.
3600 = 2^4 * 3^2 * 5^2. For a perfect cube, exponents must be multiples of 3. We have extra 2^1 * 3^2 * 5^2 = 2 * 9 * 25 = 450. Dividing by 450 gives 8 = 2^3.
How many two-digit numbers are divisible by 6?
Two-digit numbers divisible by 6: 12, 18, 24, ..., 96. This is an AP with a=12, d=6, l=96. 96 = 12 + (n-1)*6 => 84 = (n-1)*6 => n-1 = 14 => n = 15.
Simplify: (0.3 * 0.3 + 2 * 0.3 * 0.7 + 0.7 * 0.7).
Expression is in the form a^2 + 2ab + b^2 = (a + b)^2 where a = 0.3 and b = 0.7. So (0.3 + 0.7)^2 = (1.0)^2 = 1.00.
Find the LCM of 18, 24, and 36.
- Write the prime factors first: 18 = 2 × 3², 24 = 2³ × 3, and 36 = 2² × 3².
- Take the highest power of each prime: 2³ × 3² = 8 × 9 = 72.
- Check by division: 72 ÷ 18 = 4, 72 ÷ 24 = 3, and 72 ÷ 36 = 2, all whole numbers.
- Option A 54 and option C 108 both fail because neither is divisible by 24.
- Option D 144 is a common multiple of all three numbers, but the LCM must be the smallest one, and 72 is smaller.
Evaluate: 16^(3/4).
- Write the base as a power of 2: 16 = 2⁴, so 16^(3/4) becomes (2⁴)^(3/4).
- Multiply the indices: 2^(4 × 3/4) = 2³ = 8.
- A fractional index a^(m/n) means take the nth root, then raise it to the power m. The fourth root of 16 is 2, and 2 cubed is 8.
- Option A 4 is the value of 16^(1/2), and option D 12 is not a power of 2 at all.
- Converting the number to a single base before applying the index law is the quickest method in these questions.
What is the HCF of 54, 90, and 126?
54 = 18*3, 90 = 18*5, 126 = 18*7. The highest common factor among all three is 18.
Find the value of sqrt(0.000144).
0.000144 = 144 * 10^-6. Sqrt(144 * 10^-6) = 12 * 10^-3 = 0.012.
The LCM of two numbers is 12 times their HCF. The sum of HCF and LCM is 403. If one number is 93, find the other.
Let HCF = h. LCM = 12h. h + 12h = 403 => 13h = 403 => h = 31. LCM = 12 * 31 = 372. Product of numbers = HCF * LCM => 93 * x = 31 * 372 => x = (31 * 372) / 93 = 372 / 3 = 124.
How many prime numbers are there between 50 and 100?
Primes between 50 and 100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Total = 10 numbers.
Simplify: (2/3 of 3/4) / (1/2) + 1/4.
- The word "of" means multiply, so 2/3 of 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2.
- Dividing by a fraction means multiplying by its reciprocal: (1/2) ÷ (1/2) = (1/2) × 2 = 1.
- Add the last term to finish: 1 + 1/4 = 1.25.
- Follow BODMAS here. Brackets come first, then the division, and the addition is done last.
- Adding 1/4 before dividing gives 1.5, which is exactly why option C is offered as a trap.
What is the total number of prime factors of 2^3 * 3^2 * 5^4?
- "Total number of prime factors" counts every prime with its repeats, so add the exponents: 3 + 2 + 4 = 9.
- Written out in full the number is 2×2×2×3×3×5×5×5×5, and counting those nine terms gives the same result.
- Option A 3 answers a different question, the number of distinct prime factors, which are 2, 3 and 5.
- Read the wording closely, because "total" means with repetition while "distinct" or "different" means without.
- A third idea often mixed up here is the number of divisors, found by (3+1)(2+1)(4+1) = 60.
The sum of three consecutive even numbers is 78. What is the smallest number?
Let numbers be x, x+2, x+4. 3x + 6 = 78 => 3x = 72 => x = 24. Smallest number is 24.
Find the smallest square number that is divisible by each of the numbers 6, 9, and 15.
LCM of 6, 9, 15: 6 = 2*3, 9 = 3^2, 15 = 3*5. LCM = 2 * 3^2 * 5 = 90. To make it a perfect square, multiply by missing prime factors (2 and 5): 90 * 2 * 5 = 900.
Evaluate: (27/64)^(-2/3).
(27/64)^(-2/3) = (64/27)^(2/3) = ((4/3)^3)^(2/3) = (4/3)^2 = 16/9.
What is the greatest common divisor (GCD) of 105, 135, and 180?
105 = 15*7, 135 = 15*9, 180 = 15*12. The greatest common divisor among all three is 15.
Simplify: (0.2 * 0.2 + 0.01) / (0.1 * 0.1).
Numerator = 0.04 + 0.01 = 0.05. Denominator = 0.01. Result = 0.05 / 0.01 = 5.
The product of two numbers is 2028 and their HCF is 13. How many such pairs of numbers are possible?
Let numbers be 13a and 13b where a, b are coprime. 13a * 13b = 2028 => 169ab = 2028 => ab = 12. Coprime pairs for ab=12 are (1, 12) and (3, 4). So 2 pairs exist.
How many two-digit numbers are divisible by 7?
Two-digit multiples of 7 range from 14 to 98. Sequence: 14, 21, ..., 98. T_n = a + (n - 1)d => 98 = 14 + (n - 1)7 => 84 = 7(n - 1) => n - 1 = 12 => n = 13.
Evaluate: 100 – [30 – {20 – (15 – 3)}].
Innermost: 15 - 3 = 12. Next: 20 - 12 = 8. Next: 30 - 8 = 22. Final: 100 - 22 = 78.
Find the total number of factors of 120.
Prime factorization: 120 = 2^3 * 3^1 * 5^1. Number of factors = (3 + 1)(1 + 1)(1 + 1) = 4 * 2 * 2 = 16.
Evaluate: 15 + 45 / 5 * 2 – 8.
BODMAS: Division first (45/5=9) => 15 + 9 * 2 - 8. Multiplication next (9*2=18) => 15 + 18 - 8 = 25.
The HCF and LCM of two numbers are 8 and 48 respectively. If one number is 24, find the other number.
Product of two numbers = HCF * LCM => 24 * x = 8 * 48 => 24x = 384 => x = 16.
The product of two consecutive even numbers is 168. What is the larger number?
Let numbers be x and x + 2. x(x + 2) = 168 => x^2 + 2x - 168 = 0. Factors of 168 with difference 2 are 12 and 14. Larger number is 14.
Find the LCM of 14, 21, and 28.
14 = 2*7, 21 = 3*7, 28 = 2^2*7. LCM = 2^2 * 3 * 7 = 84.
Find the value of (8^(2/3)).
8 = 2^3. So (2^3)^(2/3) = 2^(3 * 2/3) = 2^2 = 4.
What is the HCF of 48, 72, and 108?
48 = 12*4, 72 = 12*6, 108 = 12*9. Highest common factor among all three is 12.
Evaluate: (100 – 1)(100 – 2)…(100 – 200).
The series includes the term (100 - 100) = 0. Multiplying any number by 0 gives 0.
Find the value of sqrt(0.0081).
0.0081 = 81 * 10^-4. Sqrt(81 * 10^-4) = 9 * 10^-2 = 0.09.
The product of two numbers is 180 and their LCM is 60. What is their HCF?
HCF * LCM = Product of numbers => HCF * 60 = 180 => HCF = 180 / 60 = 3.
How many prime numbers are there between 20 and 40?
Prime numbers between 20 and 40 are 23, 29, 31, 37. Total = 4 numbers.
Evaluate: 48 / 6 * (3 + 1) – 5.
BODMAS: Parentheses first (3+1=4) => 48 / 6 * 4 - 5. Left-to-right for / and *: (48/6)=8, 8*4=32, 32-5=27.
Find the value of (2^4 * 3^2) / 12.
(16 * 9) / 12 = 144 / 12 = 12.
Find the value of sqrt(176 + sqrt(2401)).
Sqrt(2401) = 49. Expression becomes sqrt(176 + 49) = sqrt(225) = 15.
Find the HCF of 36, 54, and 90.
36 = 18 * 2, 54 = 18 * 3, 90 = 18 * 5. Highest common factor among all three numbers is 18.
Find the unit digit of 3^45.
Cyclicity of 3 is 4 (3^1=3, 3^2=9, 3^3=7, 3^4=1). Remainder of 45 / 4 is 1. So unit digit is 3^1 = 3.
Convert 0.375 into a fraction in its simplest form.
0.375 = 375 / 1000. Dividing numerator and denominator by 125 gives 3 / 8.
Evaluate: 12 + 4 * (8 – 3) / 2.
Using BODMAS: Parentheses first (8-3=5) => 12 + 4 * 5 / 2. Division/Multiplication left-to-right => 4 * 5 = 20, 20 / 2 = 10 => 12 + 10 = 22.
What is the LCM of 12, 18, and 24?
Prime factorizations: 12 = 2^2 * 3, 18 = 2 * 3^2, 24 = 2^3 * 3. LCM = 2^3 * 3^2 = 8 * 9 = 72.
Find the value of (100^2 – 99^2).
Using identity a^2 - b^2 = (a - b)(a + b): (100 - 99)(100 + 99) = 1 * 199 = 199.
The sum of the digits of a two-digit number is 9. If 27 is added to the number, the digits are reversed. What is the number?
Let the number be 10x + y. x + y = 9. (10x + y) + 27 = 10y + x => 9y - 9x = 27 => y - x = 3. Solving: x = 3, y = 6. Number is 36.
What is the HCF of 2/3, 4/5, and 6/7?
HCF of fractions = HCF of Numerators / LCM of Denominators = HCF(2, 4, 6) / LCM(3, 5, 7) = 2 / 105.
Find the LCM of 15, 20, and 30.
Prime factorizations: 15 = 3*5, 20 = 2^2*5, 30 = 2*3*5. LCM = 2^2 * 3 * 5 = 60.
The product of two numbers is 216 and their HCF is 6. What is their LCM?
Product of two numbers = HCF × LCM, so 216 = 6 × LCM and LCM = 216 ÷ 6 = 36.
Check: 12 and 18 have HCF 6, LCM 36 and product 216.
Find the value of (1000)^0 + (100)^0 + (10)^0.
Any non-zero base with exponent 0 is 1. So 1 + 1 + 1 = 3.
Evaluate: 50 – 5 * 0 + 2 / 2.
Using BODMAS rule: Division first (2/2 = 1), then Multiplication (5*0 = 0). Expression becomes: 50 - 0 + 1 = 51.
Find the sum of first 20 natural numbers.
Formula for sum of first n natural numbers = n(n + 1) / 2 = 20 * 21 / 2 = 210.
What is 0.05 expressed as a fraction in lowest terms?
0.05 = 5 / 100 = 1 / 20.
What is the simplified value of (0.2 * 0.2 + 0.2 * 0.02) / 0.04?
Numerator = 0.04 + 0.004 = 0.044. Denominator = 0.04. Result = 0.044 / 0.04 = 1.1.
What is the square root of 0.000081?
Sqrt(81 * 10^-6) = 9 * 10^-3 = 0.009.
The sum of three consecutive integers is 135. What is the middle integer?
Let numbers be x-1, x, x+1. Sum = 3x = 135 => x = 45. The middle integer is 45.
What is the value of 2^0 + 3^0 + 4^0?
Any non-zero number raised to the power of 0 equals 1. So, 1 + 1 + 1 = 3.
The product of two consecutive odd numbers is 63. What is the larger number?
Let numbers be x and x+2. x(x+2) = 63 => x^2 + 2x - 63 = 0 => (x+9)(x-7) = 0. Positive integer x = 7. Larger number = 9.
The sum of three consecutive integers is 72. What is the smallest integer?
Let consecutive integers be x, x+1, x+2. Sum: x + (x+1) + (x+2) = 72 => 3x + 3 = 72 => 3x = 69 => x = 23.