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Basic Numeracy

295 questions

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Question 141Question →
Evaluate the following mathematical expression by applying the standard order of operations (BODMAS):
18÷[412−{214+34×(4.8−2.4+0.4‾)}]+5.518 \div \left[ 4 \frac{1}{2} - \left\{ 2 \frac{1}{4} + \frac{3}{4} \times \left( 4.8 - \overline{2.4 + 0.4} \right) \right\} \right] + 5.5
What is the final numerical value of the simplified expression?
Show answer & explanation

Answer: 29.5

Answer

The simplified numerical value of the given expression is 29.5.
Following the BODMAS rule systematically, we first resolve the vinculum to get 2.8, then evaluate the round brackets (4.8 - 2.8 = 2.0). Next, inside the curly brackets, multiplication yields 1.5 and addition yields 3.75. Evaluating the square brackets yields 4.5 - 3.75 = 0.75. Dividing 18 by 0.75 gives 24, and adding 5.5 gives the final correct answer of 29.5.

Step-by-Step Solution

1
Evaluate the expression under the vinculum bar
\overline{2.4 + 0.4} = 2.8
The vinculum (bar) takes highest priority over standard parentheses and operations.
2
Simplify the innermost round brackets
4.8 - 2.8 = 2.0
Perform subtraction inside the round brackets next.
3
Perform multiplication inside the curly brackets
34×2.0=1.5\frac{3}{4} \times 2.0 = 1.5
Multiplication precedes addition within the curly brackets according to BODMAS.
4
Complete evaluation inside the curly brackets
2 \frac{1}{4} + 1.5 = 2.25 + 1.5 = 3.75
Convert the mixed fraction 2 1/4 to decimal 2.25 and add 1.5.
5
Evaluate the terms inside the square brackets
4 \frac{1}{2} - 3.75 = 4.5 - 3.75 = 0.75
Convert mixed fraction 4 1/2 to decimal 4.5 and subtract 3.75.
6
Perform division before final addition
18÷0.75=2418 \div 0.75 = 24
Division takes precedence over the final addition operation.
7
Add the final constant
24 + 5.5 = 29.5
Complete the evaluation by adding 5.5 to 24.

Key Concept

Hierarchy of operations in BODMAS including vinculum bar, nested brackets, and precedence of operations.
Estimated Time:1m 30s
Question 142Question →
Calculate the numerical value of the following mathematical expression:
75% of 240−[42÷{214+(3.5−1.8−0.8‾)×1.5}]75\% \text{ of } 240 - \left[ 42 \div \left\{ 2\frac{1}{4} + \left( 3.5 - \overline{1.8 - 0.8} \right) \times 1.5 \right\} \right]
Show answer & explanation

Answer: 173

Answer

The simplified numerical value of the expression is 173.
Applying the VBODMAS rule sequentially (vinculum, round brackets, multiplication, addition within curly brackets, square bracket division, percentage evaluation, and final subtraction) yields 173.

Step-by-Step Solution

1
Evaluate the vinculum bar expression
1.8−0.8‾=1.0\overline{1.8 - 0.8} = 1.0
Under VBODMAS rules, the vinculum (bar) grouping has the highest priority and must be evaluated first.
2
Evaluate the subtraction inside the round brackets
3.5−1.0=2.53.5 - 1.0 = 2.5
Evaluate the parentheses next using the result from the vinculum evaluation.
3
Perform multiplication inside the curly brackets
2.5×1.5=3.752.5 \times 1.5 = 3.75
Within the curly brackets, multiplication precedes addition according to BODMAS.
4
Perform addition inside the curly brackets
214+3.75=2.25+3.75=6.02\frac{1}{4} + 3.75 = 2.25 + 3.75 = 6.0
Convert the mixed fraction 2142\frac{1}{4} to decimal 2.252.25 and add to 3.753.75.
5
Perform division inside the square brackets
42÷6.0=742 \div 6.0 = 7
Resolve the square brackets by dividing 42 by the value of the curly bracket expression.
6
Evaluate the percentage 'of' expression
75% of 240=0.75×240=18075\% \text{ of } 240 = 0.75 \times 240 = 180
The 'of' operator indicates multiplication applied to percentage.
7
Subtract the square bracket result from the percentage value
180−7=173180 - 7 = 173
Perform final subtraction to complete the calculation.

Key Concept

Hierarchical priority of operations in VBODMAS (Vinculum -> Brackets -> Of -> Division -> Multiplication -> Addition -> Subtraction)
Question 143Question →
What is the simplified numerical value of the following mathematical expression when evaluated using the standard BODMAS rule?
60% of 350−[514+{36÷(4.5−2.1+0.9‾)}]×460\% \text{ of } 350 - \left[ 5 \frac{1}{4} + \left\{ 36 \div \left( 4.5 - \overline{2.1 + 0.9} \right) \right\} \right] \times 4
Show answer & explanation

Answer: 93

Answer

The simplified numerical value of the given expression is 93.
Following the BODMAS rule: first evaluate the vinculum \(\overline{2.1 + 0.9} = 3\), then the round bracket \(4.5 - 3 = 1.5\), then the curly bracket \(36 \div 1.5 = 24\), then the square bracket \(5.25 + 24 = 29.25\). Next, evaluate the percentage term \(60\% \text{ of } 350 = 210\) and the product \(29.25 \times 4 = 117\). Finally, subtracting 117 from 210 gives 93.

Step-by-Step Solution

1
Evaluate the expression under the vinculum bar
\overline{2.1 + 0.9} = 3.0
The vinculum (bar) has the highest priority inside nested grouping symbols.
2
Evaluate the terms inside the round brackets
4.5 - 3.0 = 1.5
Subtract the result of the vinculum from 4.5 inside the round brackets.
3
Evaluate the division inside the curly brackets
36÷1.5=2436 \div 1.5 = 24
Perform division within the curly braces.
4
Evaluate the addition inside the square brackets
5 \frac{1}{4} + 24 = 5.25 + 24 = 29.25
Convert the mixed fraction to decimal (5.25) and add to 24.
5
Calculate the percentage term and multiplication term
60\% \text{ of } 350 = \frac{60}{100} \times 350 = 210 \quad \text{and} \quad 29.25 \times 4 = 117
According to BODMAS, percentage/off and multiplication must be computed before final subtraction.
6
Perform the final subtraction
210 - 117 = 93
Subtract the multiplied bracket result from the percentage value.

Key Concept

Order of Operations (BODMAS/PEMDAS) with Vinculum and Nested Brackets
Question 144Question →

What is the unit digit of the expression E=138124+26493−17754E = 138^{124} + 264^{93} - 177^{54}?

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Answer: 1

Answer

The unit digit of the expression is 1.
The correct answer is derived by determining the cyclicity of each base's unit digit: 8 has cyclicity 4 (with remainder 0 yielding unit digit 6), 4 has cyclicity 2 (odd exponent yielding unit digit 4), and 7 has cyclicity 4 (remainder 2 yielding unit digit 9). Combining these yields (6 + 4) - 9 = 1.

Step-by-Step Solution

1
Find the unit digit of 138124138^{124}
Unit digit is 6
The unit digit of the base is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 124 by 4 gives a remainder of 0. A remainder of 0 corresponds to the 4th power in the cycle (84=40968^4 = 4096), so the unit digit is 6.
2
Find the unit digit of 26493264^{93}
Unit digit is 4
The unit digit of the base is 4. The cyclicity of 4 is 2 (41=4,42=64^1 = 4, 4^2 = 6). Since the exponent 93 is odd, the unit digit is 4.
3
Find the unit digit of 17754177^{54}
Unit digit is 9
The unit digit of the base is 7. The cyclicity of 7 is 4 (7, 9, 3, 1). Dividing the exponent 54 by 4 gives a remainder of 2 (54=4×13+254 = 4 \times 13 + 2). Thus, the unit digit is 72=497^2 = 49, which ends in 9.
4
Combine the unit digits according to the expression
Unit digit is 1
Evaluating (6+4)−9=10−9=1(6 + 4) - 9 = 10 - 9 = 1. The unit digit of the entire expression is 1.

Key Concept

Unit Digit Cyclicity and Modular Exponentiation Rules
Question 145Question →

What is the unit digit of the composite exponential expression E=332520×553213−448631E = 332^{520} \times 553^{213} - 448^{631}?

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Answer: 6

Answer

The unit digit of the given expression is 6.
Evaluating each component using cyclicity of unit digits: 332520332^{520} has base unit digit 2 and exponent divisible by 4 (520 mod 4=0520 \bmod 4 = 0), so its unit digit is 24  ⟹  62^4 \implies 6. 553213553^{213} has base unit digit 3 with 213 mod 4=1213 \bmod 4 = 1, giving unit digit 31=33^1 = 3. 448631448^{631} has base unit digit 8 with 631 mod 4=3631 \bmod 4 = 3, giving unit digit 83  ⟹  28^3 \implies 2. Combining these gives (6×3)−2=18−2=16(6 \times 3) - 2 = 18 - 2 = 16, which yields a final unit digit of 6.

Step-by-Step Solution

1
Determine the unit digit of 332520332^{520} using cyclicity of 2
Unit digit is 6
The unit digit of base 332 is 2. The cyclicity of 2 is 4 (2, 4, 8, 6). Dividing the exponent 520 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power: 24=162^4 = 16, so the unit digit is 6.
2
Determine the unit digit of 553213553^{213} using cyclicity of 3
Unit digit is 3
The unit digit of base 553 is 3. The cyclicity of 3 is 4 (3, 9, 7, 1). Dividing the exponent 213 by 4 gives a remainder of 1 (213=4×53+1213 = 4 \times 53 + 1). Thus, the unit digit is 31=33^1 = 3.
3
Determine the unit digit of 448631448^{631} using cyclicity of 8
Unit digit is 2
The unit digit of base 448 is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 631 by 4 gives a remainder of 3 (631=4×157+3631 = 4 \times 157 + 3). Thus, the unit digit is 83=5128^3 = 512, which ends in 2.
4
Combine the unit digits according to the expression E=(332520×553213)−448631E = (332^{520} \times 553^{213}) - 448^{631}
Unit digit is 6
First multiply the unit digits of the first two terms: 6×3=18  ⟹  86 \times 3 = 18 \implies 8. Then subtract the unit digit of the third term: 8−2=68 - 2 = 6.

Key Concept

Unit Digit Cyclicity Rules
Estimated Time:1m 30s
Question 146Question →

Determine the exact result of the given mathematical expression after applying the standard order of operations (BODMAS):

32.4−45×[45÷{2.5×4−(7.6−4.1−1.5‾)}]32.4 - \frac{4}{5} \times \left[ 45 \div \left\{ 2.5 \times 4 - \left( 7.6 - \overline{4.1 - 1.5} \right) \right\} \right]
Show answer & explanation

Answer: 25.2

Answer

The simplified numerical value of the expression is 25.2.
Following the standard BODMAS priority order: evaluate the vinculum (4.1 - 1.5 = 2.6), round brackets (7.6 - 2.6 = 5), curly brackets (2.5 * 4 - 5 = 5), square brackets (45 / 5 = 9), multiplication ((4/5) * 9 = 7.2), and finally subtraction (32.4 - 7.2 = 25.2).

Step-by-Step Solution

1
Evaluate the sub-expression under the vinculum bar
\overline{4.1 - 1.5} = 2.6
The vinculum acts as the innermost grouping symbol and takes highest priority.
2
Simplify the expression inside the round brackets
(7.6 - 2.6) = 5.0
Parentheses (round brackets) are evaluated next.
3
Evaluate operations inside the curly brackets using operator precedence
\{2.5 \times 4 - 5.0\} = \{10 - 5.0\} = 5
Multiplication precedes subtraction inside the curly brackets.
4
Evaluate the division inside the square brackets
[45÷5]=9[45 \div 5] = 9
Square brackets are solved after resolving inner curly brackets.
5
Perform multiplication outside the bracket
\frac{4}{5} \times 9 = 0.8 \times 9 = 7.2
Multiplication takes precedence over final addition/subtraction.
6
Perform the final subtraction
32.4 - 7.2 = 25.2
Subtraction is the final step in the BODMAS order of operations.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets
Estimated Time:1m 30s
Question 147Question →

What is the unit digit of the expression K=(23385×43794)−16847K = (233^{85} \times 437^{94}) - 168^{47}?

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Answer: 5

Answer

The unit digit of the given expression is 5.
To find the unit digit of K=(23385×43794)−16847K = (233^{85} \times 437^{94}) - 168^{47}, analyze each term using base cyclicity rules:
1. 23385233^{85}: Base digit 3 has cyclicity 4. 85÷4=2185 \div 4 = 21 remainder 1. 31=33^1 = 3.
2. 43794437^{94}: Base digit 7 has cyclicity 4. 94÷4=2394 \div 4 = 23 remainder 2. 72=49→97^2 = 49 \rightarrow 9.
3. Product unit digit: 3×9=27→73 \times 9 = 27 \rightarrow 7.
4. 16847168^{47}: Base digit 8 has cyclicity 4. 47÷4=1147 \div 4 = 11 remainder 3. 83=512→28^3 = 512 \rightarrow 2.
5. Subtracting gives 7−2=57 - 2 = 5.

Step-by-Step Solution

1
Find the unit digit of 23385233^{85}
Unit digit is 3
The unit digit of powers of 3 follows a cycle of 4: (3, 9, 7, 1). Since 85≡1(mod4)85 \equiv 1 \pmod 4, the unit digit is 31=33^1 = 3.
2
Find the unit digit of 43794437^{94}
Unit digit is 9
The unit digit of powers of 7 follows a cycle of 4: (7, 9, 3, 1). Since 94≡2(mod4)94 \equiv 2 \pmod 4, the unit digit is 72=49  ⟹  97^2 = 49 \implies 9.
3
Find the unit digit of the product (23385×43794)(233^{85} \times 437^{94})
Unit digit is 7
Multiplying the unit digits of the two terms gives 3×9=273 \times 9 = 27, which has a unit digit of 7.
4
Find the unit digit of 16847168^{47}
Unit digit is 2
The unit digit of powers of 8 follows a cycle of 4: (8, 4, 2, 6). Since 47≡3(mod4)47 \equiv 3 \pmod 4, the unit digit is 83=512  ⟹  28^3 = 512 \implies 2.
5
Calculate the final unit digit of the composite expression
5
Subtracting the unit digit of the subtracted term from the product's unit digit yields 7−2=57 - 2 = 5.

Key Concept

Unit Digit and Cyclicity
Estimated Time:1m 30s
Question 148Question →

Determine the unit digit of the finite series and exponent tower expression S=∑k=150(k!)k!+777S = \sum_{k=1}^{50} (k!)^{k!} + 7^{7^7}.

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Answer: 0

Answer

The unit digit of the expression is 0.
The sum of the factorial terms from k=1 to k=4 contributes unit digits of 1, 4, 6, and 6, totaling 17 (unit digit 7). Terms for k >= 5 end in 0. The power tower term 7^(7^7) has an exponent 7^7 which leaves a remainder of 3 when divided by 4. Since the cyclicity of 7 is 4, 7^3 gives a unit digit of 3. Adding 7 and 3 results in 10, giving a final unit digit of 0.

Step-by-Step Solution

1
Calculate unit digits of factorial terms for k from 1 to 4
Unit digits are 1, 4, 6, and 6 respectively
1! = 1, 2! = 2, 3! = 6, 4! = 24 (with base ending in 4 raised to an even power 24 giving unit digit 6)
2
Analyze factorial terms for k >= 5
Unit digit is 0 for all k >= 5
Factorials for k >= 5 contain factors 2 and 5, making the trailing digit 0
3
Sum the unit digits of the series sum
Sum of unit digits is 1 + 4 + 6 + 6 = 17, giving unit digit 7
Only terms from k = 1 to 4 contribute to the unit digit of the factorial sum
4
Find the unit digit of the power tower 7^(7^7) using cyclicity modulo 4
7^7 mod 4 = 3, so unit digit is 7^3 mod 10 = 3
The base 7 has a cyclicity of 4, and 7^7 is congruent to 3 modulo 4
5
Add the unit digits of both components
7 + 3 = 10, unit digit is 0
Combining the unit digit of the factorial sum (7) and the tower exponent term (3)

Key Concept

Combining factorial unit digit termination properties with exponent tower cyclicity modulo 4.
Question 149Question →

What value is obtained upon simplifying the mathematical expression given below using the correct order of operations (BODMAS)?

87.5% of 120−[225×{24.5−(4.5÷2.4−0.9‾×1.6+115)}]87.5\% \text{ of } 120 - \left[ 2 \frac{2}{5} \times \left\{ 24.5 - \left( 4.5 \div \overline{2.4 - 0.9} \times 1.6 + 1 \frac{1}{5} \right) \right\} \right]
Show answer & explanation

Answer: 60.660.6

Answer

60.660.6
The correct result 60.660.6 is achieved by systematically resolving operations from the innermost group outward: first the bar/vinculum, followed by division and multiplication inside the parentheses, subtraction in the curly brackets, multiplication in the square brackets, and finally subtracting this total from 87.5%87.5\% of 120120.

Step-by-Step Solution

1
Evaluate the expression under the vinculum (bar)
2.4−0.9‾=1.5\overline{2.4 - 0.9} = 1.5
The vinculum has the highest priority among grouping symbols.
2
Evaluate the terms inside the parentheses (...)(...) using BODMAS
4.5÷1.5×1.6+115=3×1.6+1.2=4.8+1.2=6.04.5 \div 1.5 \times 1.6 + 1 \frac{1}{5} = 3 \times 1.6 + 1.2 = 4.8 + 1.2 = 6.0
Perform division first (4.5÷1.5=34.5 \div 1.5 = 3), then multiplication (3×1.6=4.83 \times 1.6 = 4.8), and finally addition (4.8+1.2=6.04.8 + 1.2 = 6.0).
3
Simplify the expression inside the curly brackets {...}\{...\}
24.5−6.0=18.524.5 - 6.0 = 18.5
Subtract the result of the parentheses from 24.524.5.
4
Simplify the expression inside the square brackets [...][...]
225×18.5=2.4×18.5=44.42 \frac{2}{5} \times 18.5 = 2.4 \times 18.5 = 44.4
Convert the mixed fraction 2252 \frac{2}{5} to decimal 2.42.4 and multiply by 18.518.5.
5
Calculate the percentage value
87.5% of 120=78×120=10587.5\% \text{ of } 120 = \frac{7}{8} \times 120 = 105
Convert 87.5%87.5\% to the equivalent fraction 78\frac{7}{8} and multiply by 120120.
6
Perform the final subtraction
105−44.4=60.6105 - 44.4 = 60.6
Subtract the value of the square brackets from the percentage result.

Key Concept

Strict evaluation of nested brackets (vinculum, round, curly, square) and mixed operators (percentage, fraction, division, multiplication) using the BODMAS rule.
Question 150Question →

What is the final numerical value of the following expression evaluated using the standard BODMAS rule?

50−[18÷3×(4+2)]50 - [18 \div 3 \times (4 + 2)]
Show answer & explanation

Answer: 14

Answer

The final simplified value of the expression is 14.
Following the BODMAS order of operations: first simplify the inner bracket (4+2)=6(4 + 2) = 6. Next, evaluate inside the square bracket from left to right: 18÷3=618 \div 3 = 6, and then 6×6=366 \times 6 = 36. Finally, subtract from 50 to get 50−36=1450 - 36 = 14.

Step-by-Step Solution

1
Evaluate the innermost round brackets
4+2=64 + 2 = 6
According to BODMAS, operations inside brackets must be performed first.
2
Perform division inside the square brackets
18÷3=618 \div 3 = 6
Division and multiplication have equal priority and are performed from left to right.
3
Perform multiplication inside the square brackets
6×6=366 \times 6 = 36
Multiply the quotient obtained from division by the bracketed sum.
4
Subtract the result inside brackets from 50
50−36=1450 - 36 = 14
Perform subtraction as the final operation.

Key Concept

BODMAS / Order of Operations
Question 151Question →

What is the unit digit of the composite expression E=(1!+2!+3!+⋯+20!)2026+(31×32×33×⋯×320)E = (1! + 2! + 3! + \dots + 20!)^{2026} + (3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20})?

Show answer & explanation

Answer: 8

Answer

The unit digit of the expression is 8.
Evaluating the expression requires breaking it down into two components. First, for the factorial sum 1!+2!+3!+⋯+20!1! + 2! + 3! + \dots + 20!, every term from 5!5! onward contains factors of both 2 and 5, so its unit digit is 0. The unit digit of the sum is determined solely by 1!+2!+3!+4!=331! + 2! + 3! + 4! = 33, which has a unit digit of 3. Raising 3 to the power 2026 gives 320263^{2026}. Since the unit digits of powers of 3 repeat in cycles of 4 (3, 9, 7, 1) and 2026≡2(mod4)2026 \equiv 2 \pmod 4, the unit digit of 320263^{2026} is 32=93^2 = 9.

Second, the product 31×32×⋯×3203^1 \times 3^2 \times \dots \times 3^{20} simplifies using the exponent addition rule to 31+2+⋯+20=32103^{1+2+\dots+20} = 3^{210}. Dividing 210 by 4 leaves a remainder of 2, so 32103^{210} also has a unit digit of 32=93^2 = 9.

Adding the unit digits of both terms gives 9+9=189 + 9 = 18, resulting in a final unit digit of 8.

Step-by-Step Solution

1
Find the unit digit of the inner factorial sum S=1!+2!+3!+⋯+20!S = 1! + 2! + 3! + \dots + 20!.
The unit digit of SS is 3.
For all k≥5k \ge 5, k!k! is divisible by 10 and ends in 0. Thus, only the sum of the first four terms 1!+2!+3!+4!=1+2+6+24=331! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33 determines the unit digit.
2
Calculate the unit digit of the first term S2026≡32026S^{2026} \equiv 3^{2026}.
The unit digit of the first term is 9.
The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent 2026 by 4 gives a remainder of 2 (2026=4×506+22026 = 4 \times 506 + 2). Therefore, the unit digit is 32=93^2 = 9.
3
Simplify the exponential product P=31×32×33×⋯×320P = 3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20}.
The product simplifies to 32103^{210}.
By exponent multiplication rules, 31×32×⋯×320=3∑i=120i3^1 \times 3^2 \times \dots \times 3^{20} = 3^{\sum_{i=1}^{20} i}. The sum of the first 20 positive integers is 20×212=210\frac{20 \times 21}{2} = 210.
4
Calculate the unit digit of 32103^{210}.
The unit digit of the second term is 9.
Dividing the exponent 210 by 4 gives a remainder of 2 (210=4×52+2210 = 4 \times 52 + 2). Therefore, the unit digit is 32=93^2 = 9.
5
Combine the unit digits of the two terms.
The unit digit of EE is 8.
Adding the unit digits gives 9+9=189 + 9 = 18. The unit digit of 18 is 8.

Key Concept

Unit Digit and Cyclicity of Factorial and Exponential Expressions
Estimated Time:2m 0s
Question 152Question →
What exact numerical result is obtained when simplifying the following multi-bracket expression according to the BODMAS order of operations?
75% of 240−[38.5+{1512−(6.8×2.5−4.2+3.8‾)÷0.5}]75\% \text{ of } 240 - \left[ 38.5 + \left\{ 15\frac{1}{2} - \left( 6.8 \times 2.5 - \overline{4.2 + 3.8} \right) \div 0.5 \right\} \right]
Show answer & explanation

Answer: 144

Answer

The final simplified numerical value of the given expression is 144.
Following the BODMAS rule strictly: first evaluate the vinculum 4.2+3.8‾=8\overline{4.2 + 3.8} = 8; then resolve round brackets (6.8×2.5−8=9)(6.8 \times 2.5 - 8 = 9); next resolve curly brackets (15.5−9÷0.5=−2.5)(15.5 - 9 \div 0.5 = -2.5); then square brackets (38.5−2.5=36)(38.5 - 2.5 = 36); and finally compute 75% of 240=18075\% \text{ of } 240 = 180. Subtracting 3636 from 180180 gives the correct result of 144.

Step-by-Step Solution

1
Evaluate the vinculum (bar) operator first
\overline{4.2 + 3.8} = 8
According to BODMAS, the vinculum takes precedence over round brackets.
2
Simplify the terms inside the round brackets (parentheses)
6.8 \times 2.5 - 8 = 17 - 8 = 9
Multiplication precedes subtraction inside the innermost round brackets.
3
Simplify the terms inside the curly brackets (braces)
15.5 - (9 \div 0.5) = 15.5 - 18 = -2.5
Division (9÷0.5=189 \div 0.5 = 18) is performed before subtracting from 15.515.5 (151215\frac{1}{2}).
4
Simplify the terms inside the square brackets
38.5 + (-2.5) = 36
Adding the negative result from the curly brackets to 38.538.5 yields 3636.
5
Calculate the percentage 'Of' operation and perform final subtraction
75\% \text{ of } 240 - 36 = 180 - 36 = 144
'Of' operation (75%×240=18075\% \times 240 = 180) is calculated before subtracting the bracketed total.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets
Estimated Time:2m 0s
Question 153Question →
What is the result when the mathematical expression 45−[12+(8÷2×3)]45 - [12 + (8 \div 2 \times 3)] is simplified strictly adhering to the standard BODMAS order of operations?
Show answer & explanation

Answer: 21

Answer

21
Following the BODMAS rule, operations inside the innermost parentheses are solved first from left to right: 8÷2=48 \div 2 = 4, then 4×3=124 \times 3 = 12. Next, addition inside the square brackets gives 12+12=2412 + 12 = 24. Finally, subtracting 24 from 45 yields 21.

Step-by-Step Solution

1
Evaluate division inside the parentheses
8 ÷ 2 = 4
Division and multiplication have equal priority and are evaluated from left to right inside parentheses.
2
Evaluate multiplication inside the parentheses
4 × 3 = 12
Completing the operations within the parentheses.
3
Add the values inside the square brackets
12 + 12 = 24
Evaluating the bracketed addition.
4
Subtract from the outer term
45 - 24 = 21
Performing final subtraction outside the brackets.

Key Concept

BODMAS Rule (Order of Operations)
Estimated Time:45s
Question 154Question →

What is the unit digit of the composite exponential expression E=43345+81880−62575E = 433^{45} + 818^{80} - 625^{75}?

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Answer: 4

Answer

4
Evaluating each term individually: 43345433^{45} has unit digit 33 (since 45(mod4)=145 \pmod 4 = 1), 81880818^{80} has unit digit 66 (since 80(mod4)=080 \pmod 4 = 0, giving the 4th power unit digit 84→68^4 \rightarrow 6), and 62575625^{75} has unit digit 55. Combining these gives 3+6−5=43 + 6 - 5 = 4.

Step-by-Step Solution

1
Find the unit digit of 43345433^{45}
Unit digit is 3
The unit digit of base 433 is 3. The cyclicity of 3 is 4 (31=3,32=9,33=7,34=13^1=3, 3^2=9, 3^3=7, 3^4=1). Dividing the exponent 45 by 4 gives 45=4×11+145 = 4 \times 11 + 1 (remainder 1). Thus, the unit digit is 31=33^1 = 3.
2
Find the unit digit of 81880818^{80}
Unit digit is 6
The unit digit of base 818 is 8. The cyclicity of 8 is 4 (81=8,82=4,83=2,84=68^1=8, 8^2=4, 8^3=2, 8^4=6). Dividing exponent 80 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power, giving unit digit 66 (from 84=40968^4 = 4096).
3
Find the unit digit of 62575625^{75}
Unit digit is 5
The unit digit of base 625 is 5. Any positive integer power of a number ending in 5 always ends in 5 (cyclicity of 1).
4
Combine the unit digits of all terms
4
Substitute the individual unit digits into the expression: 3+6−5=43 + 6 - 5 = 4.

Key Concept

Unit Digit and Cyclicity Rules for Exponential Expressions
Question 155Question →

Determine the unit digit of the numerical expression N=(333444×444333)+(777888×888777)−222555N = (333^{444} \times 444^{333}) + (777^{888} \times 888^{777}) - 222^{555}.

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Answer: 4

Answer

The unit digit of the given expression is 4.
By analyzing cyclicity of each base (3,4,7,8,23, 4, 7, 8, 2), we find that 333444→1333^{444} \rightarrow 1, 444333→4444^{333} \rightarrow 4, 777888→1777^{888} \rightarrow 1, 888777→8888^{777} \rightarrow 8, and 222555→8222^{555} \rightarrow 8. Thus, the overall unit digit is (1×4)+(1×8)−8=4+8−8=4(1 \times 4) + (1 \times 8) - 8 = 4 + 8 - 8 = 4.

Step-by-Step Solution

1
Calculate unit digit of 333444×444333333^{444} \times 444^{333}
Unit digit is 4
Base 3 has cyclicity 4; 444(mod4)=0  ⟹  34→1444 \pmod 4 = 0 \implies 3^4 \rightarrow 1. Base 4 has cyclicity 2; odd exponent 333  ⟹  41→4333 \implies 4^1 \rightarrow 4. Total unit digit for term 1 = 1×4=41 \times 4 = 4.
2
Calculate unit digit of 777888×888777777^{888} \times 888^{777}
Unit digit is 8
Base 7 has cyclicity 4; 888(mod4)=0  ⟹  74→1888 \pmod 4 = 0 \implies 7^4 \rightarrow 1. Base 8 has cyclicity 4; 777(mod4)=1  ⟹  81→8777 \pmod 4 = 1 \implies 8^1 \rightarrow 8. Total unit digit for term 2 = 1×8=81 \times 8 = 8.
3
Calculate unit digit of 222555222^{555}
Unit digit is 8
Base 2 has cyclicity 4; 555(mod4)=3  ⟹  23→8555 \pmod 4 = 3 \implies 2^3 \rightarrow 8.
4
Evaluate combined expression unit digit
4
Combine term unit digits: (4+8−8)=4(4 + 8 - 8) = 4.

Key Concept

Unit Digit and Cyclicity of Numbers
Question 156Question →

What is the simplified value of the following mathematical expression evaluated using the standard BODMAS rule?

24+16÷4×2−(7−5−2‾)24 + 16 \div 4 \times 2 - (7 - \overline{5 - 2})
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Answer: 28

Answer

28
Following the standard order of operations (BODMAS), we first simplify the expression under the vinculum bar: 5−2=35 - 2 = 3. Next, we complete the subtraction inside the parentheses: 7−3=47 - 3 = 4. Then, we resolve division and multiplication from left to right: 16÷4=416 \div 4 = 4, and 4×2=84 \times 2 = 8. Finally, combining addition and subtraction gives 24+8−4=2824 + 8 - 4 = 28.

Step-by-Step Solution

1
Evaluate the expression under the vinculum (bar)
\overline{5 - 2} = 3
The vinculum takes the highest priority in bracket operations.
2
Evaluate the round brackets
(7 - 3) = 4
Operations within brackets must be solved next.
3
Perform Division and Multiplication from left to right
16 \div 4 = 4; \quad 4 \times 2 = 8
Division and multiplication share equal precedence and are performed in order from left to right.
4
Perform Addition and Subtraction from left to right
24 + 8 - 4 = 28
Final addition and subtraction operations are executed left-to-right.

Key Concept

BODMAS Rule with Vinculum (Order of Operations)
Question 157Question →

What is the unit digit of the numerical expression N=(137137×264102)+619199N = (137^{137} \times 264^{102}) + 619^{199}?

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Answer: 1

Answer

1
The unit digit of 137137137^{137} is derived from 137(mod4)=1137 \pmod 4 = 1, giving 71=77^1 = 7. The unit digit of 264102264^{102} is 6 because the exponent 102 is even. Multiplying these unit digits gives 7×6=427 \times 6 = 42, which contributes a unit digit of 2. The unit digit of 619199619^{199} is 9 because 199 is odd. Adding the unit digits yields 2+9=112 + 9 = 11, whose unit digit is 1.

Step-by-Step Solution

1
Determine the unit digit of 137137137^{137} using the cyclicity of 7.
The exponent 137 leaves a remainder of 1 when divided by 4, giving a unit digit of 71=77^1 = 7.
The unit digit pattern for powers of 7 repeats every 4 powers (7, 9, 3, 1).
2
Determine the unit digit of 264102264^{102} using the cyclicity of 4.
Since 102 is an even exponent, 4even4^{\text{even}} yields a unit digit of 6.
Powers of 4 end in 4 for odd exponents and 6 for even exponents.
3
Calculate the unit digit of the product (137137×264102)(137^{137} \times 264^{102}).
(7×6)=42(7 \times 6) = 42, which has a unit digit of 2.
The unit digit of a product depends only on the product of the unit digits of its factors.
4
Determine the unit digit of 619199619^{199} using the cyclicity of 9.
Since 199 is an odd exponent, 9odd9^{\text{odd}} yields a unit digit of 9.
Powers of 9 end in 9 for odd exponents and 1 for even exponents.
5
Sum the unit digits to find the final unit digit of expression NN.
2+9=112 + 9 = 11, which has a unit digit of 1.
The unit digit of the sum of two terms is the unit digit of the sum of their individual unit digits.

Key Concept

Unit digit determination using exponent cyclicity rules for base digits 4, 7, and 9.
Estimated Time:1m 15s
Question 158Question →

If M=7+210+7−210M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} and xx satisfies the exponential equation (43)2x−1=(116)x−4\left(\sqrt[3]{4}\right)^{2x-1} = \left(\frac{1}{16}\right)^{x-4} with (43)2x−1=2k\left(\sqrt[3]{4}\right)^{2x-1} = 2^k, what is the value of M2+kM^2 + k?

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Answer: 23.5

Answer

The value of M2+kM^2 + k is 23.5.
By writing 7±2107 \pm 2\sqrt{10} as (5±2)2(\sqrt{5} \pm \sqrt{2})^2, the radical simplifies cleanly to M=25M = 2\sqrt{5}, giving M2=20M^2 = 20. Rewriting the index equation in terms of base 2 yields 4x−23=−4x+16\frac{4x-2}{3} = -4x + 16, which gives x=258x = \frac{25}{8} and exponent k=3.5k = 3.5. Adding M2M^2 and kk results in 23.5.

Step-by-Step Solution

1
Simplify the nested surd expression for M
M = 2\sqrt{5}, so M^2 = 20
Recognize that 7±210=(5±2)27 \pm 2\sqrt{10} = (\sqrt{5} \pm \sqrt{2})^2.
2
Convert both sides of the exponential equation to base 2
24x−23=2−4x+162^{\frac{4x-2}{3}} = 2^{-4x+16}
Apply laws of indices: 43=22/3\sqrt[3]{4} = 2^{2/3} and 116=2−4\frac{1}{16} = 2^{-4}.
3
Solve for x by equating the powers of 2
x=258=3.125x = \frac{25}{8} = 3.125
Since bases are equal, the powers must be equal.
4
Determine the exponent value k
k = 3.5
Substitute x into the exponent expression k=−4x+16k = -4x + 16.
5
Calculate the final combined expression M^2 + k
23.5
Add M2=20M^2 = 20 and k=3.5k = 3.5.

Key Concept

Nested radical simplification using binomial square expansion combined with solving exponential equations via prime base unification.
Question 159Question →

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digit integers PP and QQ (where P>QP > Q) are 2424 and 10801080, respectively. If the difference between the two numbers is 9696, what is the sum of the two numbers (P+QP + Q)?

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Answer: 336

Answer

The sum of the two numbers P and Q is 336.
By representing P=24aP = 24a and Q=24bQ = 24b with gcd⁡(a,b)=1\gcd(a, b) = 1, the relation HCF×LCM=P×Q\text{HCF} \times \text{LCM} = P \times Q yields a×b=45a \times b = 45. The coprime factor pairs of 4545 are (45,1)(45, 1) and (9,5)(9, 5). The pair (45,1)(45, 1) gives 10801080 and 2424, which are not both three-digit numbers. The pair (9,5)(9, 5) gives P=216P = 216 and Q=120Q = 120, both of which are three-digit numbers with a difference of 9696. The sum of these two numbers is 216+120=336216 + 120 = 336.

Step-by-Step Solution

1
Express the two numbers in terms of their HCF and coprime factors
Let P=24aP = 24a and Q=24bQ = 24b where gcd⁡(a,b)=1\gcd(a, b) = 1 and a>ba > b.
Any two numbers sharing an HCF of hh can be represented as h⋅ah \cdot a and h⋅bh \cdot b where aa and bb have no common prime factors.
2
Relate the product of the coprime factors to the LCM and HCF
a×b=LCMHCF=108024=45a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{1080}{24} = 45.
The product of two numbers equals the product of their HCF and LCM: (24a)(24b)=24×1080(24a)(24b) = 24 \times 1080.
3
Identify all coprime factor pairs of 45
The coprime factor pairs (a,b)(a, b) with a>ba > b are (45,1)(45, 1) and (9,5)(9, 5).
Factor pairs such as (15,3)(15, 3) are invalid because gcd⁡(15,3)=3≠1\gcd(15, 3) = 3 \neq 1.
4
Apply the three-digit integer and difference constraints to select the valid pair
For (a,b)=(9,5)(a, b) = (9, 5), P=24×9=216P = 24 \times 9 = 216 and Q=24×5=120Q = 24 \times 5 = 120. Difference = 216−120=96216 - 120 = 96.
The pair (45,1)(45, 1) yields P=1080P = 1080 (four digits) and Q=24Q = 24 (two digits), failing the three-digit criteria.
5
Calculate the sum of PP and QQ
P+Q=216+120=336P + Q = 216 + 120 = 336.
The question asks specifically for the sum P+QP + Q.

Key Concept

Properties of HCF and LCM including HCF x LCM = Product of Numbers and Prime Factor Coprimality
Question 160Question →
What is the simplified numerical value of the following mathematical expression when evaluated strictly according to the BODMAS rule?
45−[3×{12÷(1.5+2.7−1.2‾)}+2.5]×245 - \left[ 3 \times \left\{ 12 \div \left( 1.5 + \overline{2.7 - 1.2} \right) \right\} + 2.5 \right] \times 2
Show answer & explanation

Answer: 16

Answer

16
Evaluating the expression according to VBODMAS rules step-by-step: first the vinculum gives 1.5; adding inside round brackets gives 3; dividing inside curly braces gives 4; multiplying and adding inside square brackets gives 14.5; finally multiplying by 2 gives 29, and subtracting from 45 yields 16.

Step-by-Step Solution

1
Evaluate the expression under the vinculum (bar)
\overline{2.7 - 1.2} = 1.5
According to VBODMAS, operations under a vinculum must be simplified first.
2
Simplify the expression inside the round brackets (parentheses)
1.5 + 1.5 = 3
Parentheses have priority after the vinculum is resolved.
3
Evaluate the division inside the curly braces
12÷3=412 \div 3 = 4
Operations within braces are evaluated next.
4
Simplify the expression inside the square brackets
3 \times 4 + 2.5 = 12 + 2.5 = 14.5
Inside brackets, multiplication takes precedence over addition.
5
Perform final operations outside brackets using standard operator precedence
45 - 14.5 \times 2 = 45 - 29 = 16
Multiplication must be executed before subtraction.

Key Concept

Hierarchy of operations in VBODMAS (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)
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