This article highlights essential simplification and quick calculation problems that greatly benefit candidates preparing for competitive exams like CAT, SSC, Banking, defense, and Railway exams. Mastering these problems enhances individuals’ skills in simplifying expressions, solving equations, and performing fast and accurate calculations, boosting their chances of success in these exams.
Multiplication
Que 1: 157.03 × 17.004 – N ÷ 0.15 = 639.98 × 0.249
Find the approximate value of N.
(a) 410
(b) 376
(c) 343
(d) 238
Solution: 157 × 17 – N × 100/15 = 640 × 0.25
(160-3) × 17 – N × 20/3 = 640/4
(2720-51) – N × 20/3 = 160
N = 2509 × 3/20 = 376.35
Option (b) is correct
Que 2: 33.33% of 24% of 37.5% of 800
(a) 1
(b) 24
(c) 32
(d) 40.25
Solution: 1/3 × (25% – 1%) × 3/8 × 800
=(1/4 – 1%) × 100
=25-1
=24
Option (b) is correct
Que 3: N=80.052% of 1350.02 + 37.997% of 450.03 – 280.987
Find the approximate value of N.
(a) 970
(b) 750
(c) 860
(d) 912
Solution: 80% × 1350 + 38% × 450 – 281
=4/5 × 1350 + 450% × 38 – 281
=1080 + 9/2 × 38 – 281
=1080 + 171 – 281
= 1251 – 281
= 970
Option (a) is correct
Que 4: (27.997)² – (32.005)² + (23.998)² – √x =335
Find the approximate value of x.
(a) 0
(b) 1
(c) 4
(d) 13
Solution: 28² – 32² + 24² – √x =335
(28 – 32) (28 + 32) + 576 – √x =335
-240 + 576 – √x =335
336 – √x=335
x=1
Option (b) is correct
Que 5: 234.01 × \frac{k}{12.99} – 29.9% of 499.9 = 810.01 ÷ 26.98
Find the approximate value of k.
(a) 5
(b) 24
(c) 12
(d) 10
Solution: (234/13) × k – 30% × 500 = 810/27
18k – 150 = 30
k=10
Option (d) is correct
Que 6: 17\frac{2}{5}+5\frac{3}{4}+11\frac{5}{8}-2\frac{1}{4}\times 2\frac{3}{4}.
(a) 27\frac{31}{80}.
(b) 28\frac{47}{80}.
(c) 26\frac{3}{8}.
(d) 27\frac{41}{80}.
Solution: 17\frac{2}{5}+5\frac{3}{4}+11\frac{5}{8}-6\frac{3}{16}.
=28+\frac{47}{80}.
Option (b) is correct
Que 7: Find the value of 77777²
(a) 6094712629
(b) 6084261729
(c) 6058721629
(d) 6049261729
Solution: 11111² × 49
=123454321 × 49% × 100
=123454321 × (50% – 1%) × 100
=123454321 × (1/2 – 1/100) × 100
=(61727160.5 – 1234543.21) × 100
=60492617.29 × 100
=6049261729
Option (d) is correct
Que 8: Find the value of 992 × 1012 – 997 × 1004
(a) 2256
(b) 2826
(c) 2916
(d) 3106
Solution: Apply near 1000 concept
992 × 1012=1003904
997 × 1004= 1000988
1003904 – 1000988=2916
Option (c) is correct
Que 9: Find the sum of the digits of 99999²
(a) 36
(b) 45
(c) 27
(d) 42
Solution: From the Magical pattern of series 9
99999² = 9999800001
The sum of digits= 45
Option (b) is correct
Que 10: Find the sum of the digits of 97×99×101×103
(a) 18
(b) 27
(c) 36
(d) 45
Solution: (x-3)(x-1)(x+1)(x+3)
= (x² – 3²) (x² – 1²)
Here, x=100
= (100² – 9) (100² -1)
Product of numbers near 10,000
=99900009
Sum of digits=36
Option (c) is correct
Que 11: Find the value of 3007×2988×625
(a) 5415752500
(b) 5615572500
(c) 5725742500
(d) 5552172500
Solution: 3007×2988=8984916
8984916×625=8984916×(10000/16)
=5615572500
Option (b) is correct
Que 12: Find the approximate value of 80.052% of 1350.02 + 37.428% of 481 + 77% of 1428
(a) 2196
(b) 2270
(c) 2360
(d) 2252
Solution: 80% ×1350+ 37.5% ×480 +1428% ×77
=4/5×1350 +3/8 ×480+14.28% ×77×100
=1080+180+1/7×7700
=2360
Option (c) is correct
Que 13: Find the sum of the digits of 238×517+203×761×23-41×233
(a) 35
(b) 41
(c) 15
(d) 29
Solution: 238×517+203×761×23-41×233
Use the Digit Sum concept
=4×4+5×5×5-5×8
=7+8-4
=2
The digit sum of the option (D)= 2+9=11=2
Option (d) is correct
Test 1: Simplification
Que 1: The value of 2×3÷2 of 3×2÷(4+4×4÷4 of 4-4÷4×4) is:
(a) 8
(b) 1
(c) 4
(d) 2
Solution: 2×3÷6×2÷(4+4×4÷16-4÷4×4)
=2×1/2×2÷(4+16÷16-1×4)
=2÷(4+1-4)
=2
Option (d) is correct
Que 2: 2/3 of 9/15 ÷(5/4÷5/2×8/25) of (3/4×3/4÷1) is:
(a) 20/9
(b) 4/25
(c) 18/125
(d) 40/9
Solution: ⅔ of 9/15 ÷(5/4÷5/2×8/25) of (3/4×3/4÷1)
=6/15÷(5/4×2/5×8/25) of (9/16)
=6/15÷4/25 of 9/16
=6/15÷9/100
=6/15×100/9
=40/9
Option (d) is correct
Que 3: 7,77,77,777÷77 equals
(a) 1111
(b) 101001
(c) 10101
(d) 1010101
Solution: 1010101
Option (d) is correct
Que 4: If (√x+√y)=15 and (√x-√y)=3 then the value of √(xy)/4 is:
(a) 14.5
(b) 13.5
(c) 15.5
(d) 16.5
Solution: (√x+√y)²=15² | x+y+2√(xy)=225
(√x-√y)²=3² | x+y-2√(xy)=9
Subtract (i)-(ii)
4√(xy)=216
√(xy)=54 & √(xy)/4=13.5
Option (b) is correct
Que 5: If √4096=64, then the value of √40.96+√0.4096+√0.004086+√0.00004096 up to two places of decimals is:
(a) 7.09
(b) 7.10
(c) 7.11
(d) 7.12
Solution: 6.4+.64+.064+.0064=7.1104
Option (c) is correct
Que 6: Find the value of (3+2√2)ⁿ +(3-2√2)ⁿ, where n=-3?
(a) 189
(b) 180
(c) 108
(d) 198
Solution: [(3-2√2)³+(3+2√2)³]/{3²-(2√2)²}³
=198/1³
=198
Option (d) is correct
Que 7: For x=√(m+2n) & y=√(m-2n), the value of (x+y)/(x-y)=√3 then m:n is equal to.
(a) 2:√3
(b) √3:4
(c) √3:2
(d) 4:√3
Solution: {(x+y)+(x-y)}/{(x+y)-(x-y)}=(√3+1)/(√3-1)
x/y=(√3+1)/(√3-1)
x²/y²=(2+√3)/(2-√3)
(x²+y²)/(x²-y²)=4/2√3=2/√3
2m/4n=2/√3
m/n=4/√3
Option (d) is correct
Que 8: If x=5+2√6 then find the value of √x+1/√x
(a) 2√3
(b) 3√2
(c) 2√6
(d) 6√2
Solution: √x=√(√3+√2)²=√3+√2
1/√x =(√3-√2)
√x+1/√x=2√3
Option (a) is correct
Que 9: If x=4.6⁴+5.4⁴+24.84² & y=4.6²+5.4²+24.84 then x/y is equal to?
(a) 24.42
(b) 24.24
(c) 25.42
(d) 25.48
Solution: a⁴+b⁴+(ab)²=(a²+b²+ab)(a²+b²-ab)
x/y=4.6²+5.4²-4.6×5.4 =25.48
Option (d) is correct
Que 10: If (4+3√3)/√(7+4√3) =A+√B then find the value of B-A?
(a) -13
(b) 2√13
(c) 13
(d) 3√3-√7
Solution: √(7+4√3)=√(2+√3)²=2+√3
(4+3√3)(2-√3)/1= -1+2√3
A+√B=-1+√12
B-A=12-(-1)=13
Option (c) is correct
Test 2: Simplification
Que 1: If ‘+’ means ‘-’ , ‘-’ means ‘+’, ‘×’ means ‘÷’ and ‘÷’ means ‘×’ then the value of
\frac{42-12\times 3+8\div 2+15}{8\times 2-4+9\div 3}.
(a) -15/19
(b) 15/19
(c) 5/3
(d) -5/3
Solution:
(42+12÷3-8×2-15)=42+4-16-15=15
(8÷2+4-9×3)=4+4-27=-19
Option (a) is correct
Que 2: Find the value of the given expression
\sqrt{\frac{4\frac{1}{7}-2\frac{1}{4}}{3\frac{1}{2}+1\frac{1}{7}}\div\frac{1}{2+\frac{1}{2+\frac{1}{5-\frac{1}{5}}}}}.
(a) 1
(b) 4
(c) 3
(d) 2
Solution:
A=4(1/7) -2(1/4)=29/7-9/4=(116-63)/28=53/28
B=3(1/2)+1(1/7)=7/2+8/7=(65/14)
A/B=53/130
5-1/5=24/5
2+5/24=53/24
2+24/53=130/53
53/130
53/130÷ 53/130=1
Option(a) is correct.
Que 3: Find the value of the given expression
[(1+\frac{1}{10+\frac{1}{10}})^2- (1-\frac{1}{10+\frac{1}{10}})^2]\div [(1+\frac{1}{10+\frac{1}{10}})(1-\frac{1}{10+\frac{1}{10}})].
(a) 20/101
(b) 100/101
(c) 2
(d) None
Solution:
(a²-b²)÷ab={(a+b)(a-b)}÷ab
a=1+10/101=111/101
b=1-10/101=91/101
a+b=202/101
a-b=20/101
ab=(111×91)/101²
(a+b)(a-b)/ab=(202×20)/111×91=4040/10101
Option (d) is correct
Que 4: Find the value of the given expression
\frac{1.2^3+0.8^3+0.7^3-2.016}{1.35\times [1.2^2+0.8^2+0.7^2-0.96-0.84-0.56]}.
(a) 1/4
(b) 1/2
(c) 1
(d) 2
Solution:
a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)
Note: Here (a+b+c)=1.2+0.8+0.7=2.7
Option (d) is correct
Que 5: If x=\sqrt{5+\sqrt{5+\sqrt{5+…}}} and y=\sqrt{5-\sqrt{5-\sqrt{5-…}}} then which one is correct:
(a) x-y+1=0
(b) x+y-1=0
(c) x+y+1=0
(d) x-y-1=0
Solution:
x²=5+x | y²=5-y
x²-x=5 | y²+y=5
x²-x=y²+y
x²-y²=x+y
(x+y)(x-y)=(x+y)
x-y=1
x-y-1=0
Option (d) is correct
Que 6: Find the value of the given expression
(5^{\frac{1}{4}}-1)(5^{\frac{3}{4}}+5^{\frac{1}{2}}+5^{\frac{1}{4}}+1).
(a) 5
(b) 25
(c) 10
(d) 4
Solution:
(a-1) (a³+a²+a+1)=(a⁴-1)
a⁴-1=5-1=4
Option (d) is correct
Que 7: Find the value of the given expression
\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4...}}}}}}.
(a) 32
(b) 8
(c) 4
(d) 2
Solution:
(x²/2)³=4x
x6=32x
x5=32
x=2
Option (d) is correct
Que 8: \sqrt{5\sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}}}=?
(a) 5^(1/32)
(b) 5^(15/32)
(c) 5^(31/32)
(d) 5^(1/16)
Solution:
Number of roots, n=5
So, answer=5^(1-1/2ⁿ) =5^(31/32)
Option (c) is correct
Que 9: Find the value of the given expression
2\sqrt[3]{32}-3\sqrt[3]{4}+\sqrt[3]{500}.
(a) 3\sqrt{24}.
(b) 6\sqrt[3]{4}.
(c) 2\sqrt[3]{24}.
(d) None
Solution:
2×2×(4)^⅓ – 3×(4)^⅓ +5×(4)^⅓
=6×(4)^⅓
Que 10: Find the value of the given expression
[(-2)^{(-2)}]^{(-2)}.
(a) -1
(b) -8
(c) 8
(d) 16
Solution:
(-1/4)^(-2)=(-4)²=16
Test 3: Simplification
Que 1: Select the correct combination of mathematical signs that can sequentially replace the * sign and balance the given equation.
42*7*64*11*6*4
(a) ÷, -, +, x, =
(b) ×, +, -, ÷, =
(c) ÷, +, -, ×, =
(d) ×, -, +, ÷, =
Solution:
42÷7+64-11×6=6+64-66=6-2=4
Option (c) is correct
Que 2: \frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99} is equal to:
(a) 10/11
(b) 9/11
(c) 5/11
(d) 7/11
Solution: \frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\frac{1}{7\times 9}+\frac{1}{9\times 11}.
=\frac{1}{2}[\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+.....+\frac{1}{9}-\frac{1}{11}].
=\frac{1}{2}[1-\frac{1}{11}].
=5/11
Option (c) is correct
Que 3: Find the value of the given expression
\frac{0.8\overline{3}\div 7.5}{2.3\overline{21}-0.0\overline{98}}.
(a) 0.05
(b) 0.6
(c) 0.06
(d) 0.1
Solution: 0.8\overline{3}=\frac{83-8}{90}=\frac{75}{90}.
2.3\overline{21}=2+0.3\overline{21}=2+\frac{321-3}{990}=2+\frac{318}{990}.
0.0\overline{98}=\frac{98}{990}.
Final answer=0.05
Option (a) is correct
Que 4:Find the value of ‘*’ in the following
1\frac{2}{3}\div\frac{2}{7}\times\frac{*}{7}=1\frac{1}{4}\times\frac{2}{3}\div\frac{1}{6}.
(a) 6
(b) 1/6
(c) 0.006
(d) 0.6
Solution: \frac{5}{3}\times\frac{7}{2}\times\frac{*}{7}=\frac{5}{4}\times\frac{2}{3}\times 6.
*=6
Option (a) is correct
Que 5: If a=4.36, b=2.39, c=1.97 then the value of a³-b³-c³-3abc is:
(a) 3.94
(b) 2.39
(c) 0
(d) 1
Solution: a³+b³+c³-3abc=0 if (a+b+c)=0
or
a³-(-b)³-(-c)³-3a(-b)(-c)=a³-b³-c³+3abc=0
If {a+(-b)+(-c)}=(a-b-c)=0
Here, a-b-c=0 given
Option (c) is correct
Que 6: Find the value of the given expression
y-[y-(x+y)-{y-(y-\overline{x-y})}+2x].
(a) 0
(b) x
(c) 4x
(d) 2y
Solution:
y-[y-(x+y)-{y-(y-x+y)}+2x]
y-[y-(x+y)-{y-2y+x}+2x]
y-[y-(x+y)-{x-y}+2x]
y-[y-x-y-x+y+2x]
y-[y]=0
Option (a) is correct
Que 7: Find the value of the given expression
56\div[\frac{1}{3}\{15+12-(9+6-\overline{5+7})\}].
(a) 12
(b) 4032
(c) 7
(d) 448
Solution: 56\div[\frac{1}{3}\{15+12-(9+6-12)\}].
56\div[\frac{1}{3}\{15+12-3\}].
56\div 8=7.
Option (c) is correct
Que 8: Find the value of the given expression
17\frac{2}{5}+11\frac{3}{4}+5\frac{1}{5}-7\frac{1}{2}-8\frac{3}{5}+2\frac{1}{4}.
(a) 18.75
(b) 19.20
(c) 20.5
(d) 21.75
Solution: (17+11+5-7-8+2)+(2/5+1/5-3/5)+(3/4-1/2+1/4)
=20+0+0.5
=20.5
Option (c) is correct
Que 9: Find the value of the given expression
\frac{1}{\sqrt{12-\sqrt{140}}}-\frac{1}{\sqrt{8-\sqrt{60}}}-\frac{2}{\sqrt{10+\sqrt{84}}}.
(a) 0
(b) 1
(c) 2
(d) 3
Solution: \sqrt{12-\sqrt{140}}=\sqrt{7}-\sqrt{5}.
\sqrt{8-\sqrt{60}}=\sqrt{5}-\sqrt{3}.
\sqrt{10+\sqrt{84}}=\sqrt{7}+\sqrt{3}.
\frac{1}{2}[(\sqrt{7}+\sqrt{5})-(\sqrt{5}+\sqrt{3})-(\sqrt{7}-\sqrt{3})]=0.
Option (a) is correct
Que 10: Find the value of a+b
\left(\frac{p^{-1}q^{2}}{p^{3}q^{-2}}\right)^{\frac{1}{3}}\div\left(\frac{p^{6}q^{-3}}{p^{-2}q^{3}}\right)^{\frac{1}{3}}=p^{a}q^{b}.
(a) -1
(b) -2/3
(c) 1/3
(d) 0
Solution: p^{\frac{-4}{3}}q^{\frac{4}{3}}\div p^{\frac{8}{3}}q^{\frac{-6}{3}}.
p^{\left(\frac{-4}{3}-\frac{8}{3}\right)}q^{\left(\frac{4}{3}+2\right)}=p^{-4}q^{\frac{10}{3}}.
a=-4 , b=10/3
a+b=-2/3
Option (b) is correct
Classification of Numbers
Que 1: Which of the following is an integer but not a whole number?
(a) -2
(b) -\frac{1}{3}.
(c) 5
(d) 2\frac{1}{3}.
Solution: Integers={…,-3,-2,-1,0,1,2,3,….}
Whole numbers={0,1,2,3,….}
Option (a) is correct
Que 2: Which of the following does not describe the number 5?
(a) Real number
(b) Whole number
(c) Complex number
(d) None
Solution: 5 is a real, whole, Integer, and complex number with the imaginary part equal to zero.
Option (d) is correct
Que 3: Which one is not a surd?
(a) \sqrt{2}
(b) 8\sqrt{3}
(c) \sqrt{\pi}
(d) \sqrt{2\frac{3}{5}}
Solution: The square root of a rational number, having an irrational solution is called surd but 𝜋 is an irrational number.
Option (c) is correct
Que 4: Zero (0) is what type of number?
(a) Real number
(b) Imaginary number
(c) Rational number
(d) None
Solution: All of the above
Option (d) is correct
Que 5: Which one describes the set {0,1,2,3,4….}?
(a) Natural number
(b) Integers
(c) Whole numbers
(d) Rational Numbers
Solution: Option (c) is correct
Que 6: 5/8 is a ?
(a) Natural number
(b) Whole number
(c) Rational number
(d) Integer
Solution: Option (c) is correct
Que 7: Which number is not irrational?
(a) √2
(b) √4.41
(c) √22.5
(d) √𝜋
Solution: √4.41=2.1
Option (b) is correct
Que 8: Natural numbers are also called?
(a) Counting numbers
(b) Elementary numbers
(c) Real numbers
(d) None
Solution: Option (a) is correct
Que 9: Which type of numbers can’t be written in fractions?
(a) Real numbers
(b) Complex numbers
(c) Surds
(d) Zero
Solution: Surds are irrational numbers.
Option (c) is correct
Que 10: Classify the number -12?
(a) Rational, Complex, Transcendental
(b) Real, irrational, integer
(c) Rational, Integer, Real number
(d) Real, Irrational, Whole
Solution: Option (c) is correct
Properties of Prime Numbers: Click Here
Que 11: Find the remainder when 7392 is divided by 24?
(a) 58
(b) 7
(c) 1
(d) 0
Solution: P2 mod 24=1, Here Prime Number P>3
Refer to fact number:8
Option (c) is correct
Que 12: If a2-b2=103 where a and b are consecutive numbers then find the value of 2a+3b?
(a) 192
(b) 204
(c) 257
(d) 312
Solution: 103=(\frac{103+1}{2})^{2}-(\frac{103-1}{2})^{2}=52^2-51^2.
a=52 and b=51
2a+3b=257
Refer to fact number:9
Option (c) is correct
Que 13: Which number is not a prime number?
(a) 659
(b) 767
(c) 877
(d) 967
Solution: 767 is divisible by 13
Option (b) is correct
Que 14: Find the remainder when 2437 is divided by 37?
(a) 36
(b) 1
(c) 24
(d) 12
Solution: \frac{24^{37}}{37}=\frac{24^{36}\times 24}{37}.
2436×24 mod 37=24
As we know 2436 mod 37=1
Refer to Fermat’s Little Theorem.
Option (c) is correct
Que 15: Find the remainder when 15! is divided by 17?
(a) 1
(b) 8
(c) 15
(d) 16
Solution: 16! mod 17=16
16×15! mod 17=16
16×k=16
k=1 (Wilson Theorem)
Refer to Fact Number:6
Option (a) is correct
Que 16: Find the average of all Prime numbers from 1 to 100?
(a) 38.25
(b) 41.50
(c) 42.40
(d) 43.80
Solution: Option (c) is correct
Que 17: When a four-digit prime number is divided by 6 then what could be the possible remainder?
(a) 2
(b) 3
(c) 4
(d) 5
Solution: Prime number=6n±1
So, the remainder would be 1 or -1
Here -1 means 6-1=5
Option (d) is correct
Que 18: Which number is a Prime Number?
(a) 144711
(b) 144723
(c) 144757
(d) 144765
Solution: Options (a),(b), and (d) are divisible by 3
Option (c) is correct
Que 19: Which number is a composite number?
(a) 457
(b) 577
(c) 679
(d) 883
Solution: 679 is divisible by 7
Option (c) is correct
Que 20: Find the sum of the digits of the Prime number (P) if P-66 is divisible by a perfect square number?
(a) 12
(b) 16
(c) 19
(d) 25
Solution: P=x2+x+41
x+41=66
x=25
P=625+25+41=691
Digit Sum=6+9+1=16
Refer to Fact Number: 7
Option (b) is correct