Question 1

The sum of the first 20 terms of the series $\frac{1}{5 	imes 6} + \frac{1}{6 	imes 7} + \frac{1}{7 	imes 8} + \dots$ is

Accepted Answer

Concept & Formula: This is a telescoping series where each individual term can be split using partial fractions: (1/n(n+1)) = (1/n) - (1/n+1) Step-by-Step Solution: • Identify the form of the n-th term: T_n = (1/(n+4)(n+5)) • Determine the 20th term of this series: For n = 1, the term is (1/5 × 6). For n = 20, the term is (1/(20+4) × (20+5)) = (1/24 × 25). • Expand the sum of the 20 terms using the splitting method: S_{20} = ((1/5) - (1/6)) + ((1/6) - (1/7)) + \dots + ((1/24) - (1/25)) • Cancel out the intermediate terms (telescoping property): S_{20} = (1/5) - (1/25) • Compute the final fractional and decimal value: S_{20} = (5 - 1/25) = (4/25) (4/25) = 0.16 Exam Strategy & Shortcut: For any standard telescoping series with a difference of 1 in the denominator factors, the sum is always: Sum = First Factor^{-1} - Last Factor^{-1} Here, the first factor is 5, and the last factor after 20 terms is 5 + 20 = 25. (1/5) - (1/25) = (4/25) = 0.16 Common Pitfall: Mistakenly calculating the 20th term's denominator as 20 × 21 instead of tracking the initial offset starting at 5. Always verify the first term match before computing the final term. Final Answer: Therefore, the correct answer is 0.16.

Question 2

The last digit in the decimal representation of $\left(\frac{1}{5}\right)^{2000}$ is

Accepted Answer

Concept & Logic: Convert the fraction into its standard decimal base. The behavior of powers applied to decimals determines the terminating trailing digit. Step-by-Step Solution: • Express the fraction as a decimal: (1/5) = 0.2 • The given term becomes: (0.2)^{2000} = ((2/10))^{2000} = \frac{2^{2000}}{10^{2000}} • Analyze the cyclicity of the last digit of powers of 2: • 2^1 = 2 • 2^2 = 4 • 2^3 = 8 • 2^4 = 16 (ends in 6) The cyclicity length is 4. • Find the remainder of the exponent when divided by 4: 2000 \div 4 = 500 \quad (Remainder 0) A remainder of 0 corresponds to the 4th position in the cycle, meaning 2^{2000} ends exactly in the digit 6. • Dividing an integer ending in 6 by a power of 10 shifts the decimal point but leaves the final non-zero terminating rightmost digit as 6. Exam Strategy & Shortcut: Recognize that ((1/5))^n = (0.2)^n. The last digit matches the unit digit cyclicity pattern of 2^n. Because 2000 is perfectly divisible by 4, the units digit matches 2^4 = 16, which ends in 6. Common Pitfall: Students often confuse "last digit in the decimal representation" with "first decimal place", picking 5 because they think of the fraction 5 dynamically, or selecting 2 without applying the exponent cycle. Final Answer: Therefore, the correct answer is 6.

Question 3

$(78.95)^2 - (43.35)^2 = $ $x$

Accepted Answer

Concept & Formula: This problem tests your ability to simplify calculations using the difference of squares identity. a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: • Let a = 78.95 and b = 43.35. • Apply the difference of squares formula: (78.95)^2 - (43.35)^2 = (78.95 - 43.35)(78.95 + 43.35). • Calculate the difference (a - b): 78.95 - 43.35 = 35.60 • Calculate the sum (a + b): 78.95 + 43.35 = 122.30 • Multiply the two results: 35.60 × 122.30 • To simplify multiplication, you can treat it as 35.6 × 122.3 = 4353.88. Exam Strategy & Shortcut: Use the unit digit or approximation to verify your answer quickly. The sum is roughly 122 and the difference is roughly 35. 122 × 35 is approximately 4270. Looking at the exact decimals, 0.6 × 0.3 will end in an 8. Option (d) 4353.88 matches both the approximate magnitude and the final decimal digit perfectly. Common Pitfall: Squaring 78.95 and 43.35 independently. This is computationally heavy and highly error-prone under time pressure. Always look for algebraic identities first. Final Answer: Therefore, the correct answer is 4353.88.

Question 4

$$ \frac{(3.63)^2 - (2.37)^2}{3.63 + 2.37} $$  is simplified to

Accepted Answer

Concept & Formula: This is a classic simplification problem relying on the difference of squares property. The denominator is designed to cancel out the addition factor of the expanded numerator. (a^2 - b^2/a + b) = ((a - b)(a + b)/a + b) = a - b Step-by-Step Solution: • Identify the variables: a = 3.63 and b = 2.37. • The expression is in the exact format of (a^2 - b^2/a + b). • Based on the algebraic identity, the (a + b) terms in the numerator and denominator cancel each other out. • You are left with only (a - b). • Calculate the difference: 3.63 - 2.37 = 1.26 Exam Strategy & Shortcut: Do not even write out the formula. When you see (a^2 - b^2/a + b), your brain should immediately jump to calculating a - b. Perform the subtraction 3.63 - 2.37 = 1.26 mentally. Common Pitfall: A careless mistake during mental subtraction is the only real risk here (e.g., getting 1.36 instead of 1.26 by messing up the borrow/carry step in decimals). Always double-check your arithmetic! Final Answer: Therefore, the correct answer is 1.26.

Question 5

$$ \frac{(36.54)^2 - (3.46)^2}{x} = 40 $$

Accepted Answer

Concept & Formula: The equation asks us to solve for an unknown denominator by first simplifying the numerator using the difference of squares identity. a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: • Let the unknown value be x. • Focus on the numerator: (36.54)^2 - (3.46)^2. • Apply the formula where a = 36.54 and b = 3.46: (36.54 - 3.46)(36.54 + 3.46) • Calculate the sum: 36.54 + 3.46 = 40.00. • Calculate the difference: 36.54 - 3.46 = 33.08. • Substitute these back into the equation: (33.08 × 40/x) = 40 • The 40 on the left side of the numerator cancels out with the 40 on the right side of the equation. • This leaves (33.08/x) = 1 , which means x = 33.08. Exam Strategy & Shortcut: Realize that the sum of the two numbers (36.54 + 3.46) is exactly 40. Because the right-hand side of the equation is also 40, it means the (a+b) component is canceled out. Therefore, x must be exactly equal to the (a-b) component. Just calculate 36.54 - 3.46 = 33.08. Common Pitfall: Misplacing the decimal point during the subtraction step. Looking at the options, 3.308, 33.08, and 330.8 are provided specifically to trap students who misalign their decimals. Final Answer: Therefore, the correct answer is 33.08.

Question 6

$$ \left(\frac{1.49 	imes 14.9 - 0.51 	imes 5.1}{14.9 - 5.1}ight) $$ is equal to

Accepted Answer

Concept & Formula: This problem simplifies decimal arithmetic by converting numbers to a common base to reveal a hidden difference of squares identity. a^2 - b^2 = (a - b)(a + b) Step-by-Step Solution: • Notice the terms in the numerator can be rewritten by factoring out (1/10) or 0.1: • 1.49 × 14.9 = (14.9/10) × 14.9 = ((14.9)^2/10) • 0.51 × 5.1 = (5.1/10) × 5.1 = ((5.1)^2/10) • Rewrite the numerator using these common bases: ((14.9)^2 - (5.1)^2/10) • Apply the difference of squares identity a^2 - b^2 = (a - b)(a + b) to the numerator: (14.9)^2 - (5.1)^2 = (14.9 - 5.1)(14.9 + 5.1) • Substitute this back into the overall fraction: (\frac{(14.9 - 5.1)(14.9 + 5.1)/10)}{14.9 - 5.1} • The term (14.9 - 5.1) cancels out from both the numerator and denominator: (14.9 + 5.1/10) • Calculate the final sum and divide: (20/10) = 2.00 Exam Strategy & Shortcut: Look at the options and decimal placements. The numerator is roughly (1.5 × 15) - (0.5 × 5) = 22.5 - 2.5 = 20. The denominator is 14.9 - 5.1 = 9.8. Thus, the expression values to roughly (20/9.8) \approx 2. This instantly eliminates (a), (c), and (d). Common Pitfall: Trying to multiply 1.49 × 14.9 and 0.51 × 5.1 directly. This leads to tedious multi-digit decimal operations that waste valuable time. Final Answer: Therefore, the correct answer is 2.00.

Question 7

$$ \frac{(0.6)^4 - (0.5)^4}{(0.6)^2 + (0.5)^2} $$ is equal to

Accepted Answer

Concept & Formula: This expression uses a higher-power difference of squares. Any expression in the form of a^4 - b^4 can be factored as a difference of squares by treating it as (a^2)^2 - (b^2)^2. x^2 - y^2 = (x - y)(x + y) \implies a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) Step-by-Step Solution: • Express the numerator using the identity framework where a = 0.6 and b = 0.5: (0.6)^4 - (0.5)^4 = ((0.6)^2 - (0.5)^2)((0.6)^2 + (0.5)^2) • Substitute this expansion back into the original fraction format: (((0.6)^2 - (0.5)^2)((0.6)^2 + (0.5)^2)/(0.6)^2 + (0.5)^2) • Cancel the common factor ((0.6)^2 + (0.5)^2) from both the numerator and the denominator. • This leaves you with: (0.6)^2 - (0.5)^2 • Compute the squares and find the difference: 0.36 - 0.25 = 0.11 Exam Strategy & Shortcut: Recognize the identity form (a^4 - b^4/a^2 + b^2) = a^2 - b^2 automatically. Skip any intermediate writing steps and jump directly to computing (0.6)^2 - (0.5)^2 = 0.36 - 0.25 = 0.11. Common Pitfall: Selecting option (a) 0.1 by mistakenly computing 0.6 - 0.5 instead of keeping the remaining terms squared ((0.6)^2 - (0.5)^2). Final Answer: Therefore, the correct answer is 0.11.

Question 8

$(7.5 	imes 7.5 + 37.5 + 2.5 	imes 2.5)$ is equal to

Accepted Answer

Concept & Formula: This expression fits the classic quadratic expansion pattern for the square of a sum. a^2 + 2ab + b^2 = (a + b)^2 Step-by-Step Solution: • Let a = 7.5 and b = 2.5. • Map out the components of the given expression: • a^2 = 7.5 × 7.5 • b^2 = 2.5 × 2.5 • Test the middle term for 2ab: 2 × 7.5 × 2.5 = 15 × 2.5 = 37.5. • Since the middle term matches perfectly, rewrite the entire expression as (a + b)^2: (7.5 + 2.5)^2 • Simplify inside the parentheses: 7.5 + 2.5 = 10 • Square the result: 10^2 = 100 Exam Strategy & Shortcut: Whenever you see a^2 + middle term + b^2, check if the middle term is equal to 2ab. Here, 7.5 × 2.5 = 18.75, and doubling it gives 37.5. Recognizing this lets you instantly convert the expression to (7.5 + 2.5)^2 = 10^2 = 100 within seconds. Common Pitfall: Manually executing the multiplications (7.5 × 7.5 = 56.25 and 2.5 × 2.5 = 6.25) and adding them to 37.5. While this works, it increases computational overhead and the risk of basic addition mistakes. Final Answer: Therefore, the correct answer is 100.

Question 9

$(99.75)^2 - 2250.0625 =$ $x$

Accepted Answer

Concept & Formula: Rather than performing a complex multiplication for (99.75)^2, apply the algebraic identity for the square of a difference to calculate it quickly. (a - b)^2 = a^2 - 2ab + b^2 Step-by-Step Solution: • Step 1: Calculate (99.75)^2 • Rewrite 99.75 as (100 - 0.25) to make the math easier. • Apply the formula where a = 100 and b = 0.25: (100 - 0.25)^2 = (100)^2 - 2(100)(0.25) + (0.25)^2 • Solve the components: • (100)^2 = 10000 • 2 × 100 × 0.25 = 50 • (0.25)^2 = 0.0625 • Combine them: 10000 - 50 + 0.0625 = 9950.0625. • Step 2: Perform the final subtraction • Substitute the squared value back into the original equation: 9950.0625 - 2250.0625 • The decimal portions (0.0625) perfectly cancel each other out. • 9950 - 2250 = 7700. Exam Strategy & Shortcut: Look at the options and the numbers in the question. The number you are subtracting (2250.0625) ends in .0625. This strongly hints that (99.75)^2 will also end in .0625 (since 0.75^2 involves similar decimal trailing). Knowing the decimals will cancel out, focus only on estimating the integers. 100^2 is 10,000. 10,000 - 2250 is 7750. The only option close to this integer is 7700. Common Pitfall: Attempting to multiply 99.75 × 99.75 manually using long multiplication. This is incredibly time-consuming and highly prone to alignment and addition errors. Final Answer: Therefore, the correct answer is 7700.

Question 10

$(55.25)^2 - 637.5625 =$ $x$

Accepted Answer

Concept & Formula: This requires calculating the square of a decimal ending in 0.25. We can use the square of a sum identity to break down the calculation into simple integer operations. (a + b)^2 = a^2 + 2ab + b^2 Step-by-Step Solution: • Step 1: Calculate (55.25)^2 • Rewrite 55.25 as (55 + 0.25). • Apply the formula with a = 55 and b = 0.25: (55 + 0.25)^2 = (55)^2 + 2(55)(0.25) + (0.25)^2 • Find (55)^2: A number ending in 5 squared is n(n+1) prepended to 25. So, 5 × 6 = 30, making it 3025. • Find the middle term: 2 × 55 × 0.25 = 110 × 0.25 = (110/4) = 27.5. • Find the last term: (0.25)^2 = 0.0625. • Add them together: 3025 + 27.5 + 0.0625 = 3052.5625. • Step 2: Execute the final equation • Subtract the given value from our squared result: 3052.5625 - 637.5625 • The .5625 decimals perfectly cancel out, leaving a clean integer subtraction. • 3052 - 637 = 2415. Exam Strategy & Shortcut: Use fast approximation combined with decimal logic. You know (55)^2 is roughly 3000. Subtracting \approx 640 from 3000 leaves roughly 2360. Looking at the choices, 2415 is the only plausible answer. Furthermore, recognizing that (0.25)^2 generates a trailing .0625 confirms that subtracting .5625 will result in a whole number. Common Pitfall: Getting bogged down in traditional multiplication for 55.25 × 55.25. Whenever a number ends in .25 or .75, splitting it into an integer plus a fraction (like 1/4 or 3/4) makes mental math significantly faster. Final Answer: Therefore, the correct answer is 2415.

Question 11

The value of $(.98)^3 + (.02)^3 + 3 	imes .98 	imes .02 - 1$ is

Accepted Answer

Concept & Formula: This expression is a cleverly disguised application of the algebraic identity for the cube of a sum. (a + b)^3 = a^3 + b^3 + 3ab(a + b) Step-by-Step Solution: • Identify the components of the expression: (0.98)^3 + (0.02)^3 + 3 × 0.98 × 0.02 - 1. • Let a = 0.98 and b = 0.02. • Notice that the sum of these variables is exactly 1: a + b = 0.98 + 0.02 = 1 • Look at the 3ab term in the problem: 3 × 0.98 × 0.02. Since multiplying by 1 changes nothing, we can invisibly multiply this term by (a + b) because a + b = 1. • The expression (0.98)^3 + (0.02)^3 + 3(0.98)(0.02)(0.98 + 0.02) perfectly matches the expansion of (a + b)^3. • Collapse the expansion back into the cubed binomial: (0.98 + 0.02)^3 • Solve the cube: (1)^3 = 1 • The original problem asks for this entire expression minus 1: 1 - 1 = 0 Exam Strategy & Shortcut: In competitive exams, anytime you see two cubed decimal numbers that add up to exactly 1.0, accompanied by a 3 × number × number term, immediately recognize the (a+b)^3 identity. Since a+b=1, the entire first segment of the equation simply evaluates to 1^3, which is 1. The final step 1 - 1 = 0 is immediate. Common Pitfall: Failing to realize that the (a+b) term is "missing" from the equation because it equals 1. Students often try to manually cube 0.98, which is a brutal calculation that guarantees a massive loss of time. Final Answer: Therefore, the correct answer is 0.

Question 12

The value of $$ \frac{5.71 	imes 5.71 	imes 5.71 - 2.79 	imes 2.79 	imes 2.79}{5.71 	imes 5.71 + 5.71 	imes 2.79 + 2.79 	imes 2.79} $$ is

Accepted Answer

Concept & Formula: This problem is built entirely around the algebraic expansion for the difference of two cubes. The required formula is: a^3 - b^3 = (a - b)(a^2 + ab + b^2) Step-by-Step Solution: Assign variables to the repeating numbers: Let a = 5.71 and b = 2.79. Observe the structure of the given expression: • The numerator represents a^3 - b^3. • The denominator represents a^2 + ab + b^2. Convert the numerical fraction into its algebraic form: (a^3 - b^3/a^2 + ab + b^2) Substitute the expanded form of a^3 - b^3 into the numerator: ((a - b)(a^2 + ab + b^2)/a^2 + ab + b^2) The term (a^2 + ab + b^2) is common to both the numerator and the denominator, so it cancels out completely. The simplified result is simply (a - b). Finally, subtract the actual decimal values: 5.71 - 2.79 = 2.92 Exam Strategy & Shortcut: When you see a difference of cubes a^3 - b^3 divided by its corresponding trinomial a^2 + ab + b^2, simply subtract the second number from the first: a - b. By looking at the last digits (1 - 9), you know the final digit must end in a 2. This helps eliminate options quickly before even doing the full subtraction. Common Pitfall: Students sometimes confuse the signs in the formula, mistakenly adding the numbers instead of subtracting them. Option (C) 8.50 is exactly what you would get if you mistakenly computed a + b. Always match the sign of the operation between the cubed terms! Final Answer: Therefore, the correct answer is 2.92.

Question 13

$$ \left[ \frac{8 (3.75)^3 + 1}{(7.5)^2 - 6.5} \right] $$ is equal to:

Accepted Answer

Concept & Formula: This is a heavily disguised sum of cubes problem. The challenge is using the properties of exponents to manipulate the numerator and denominator into the standard a^3 + b^3 identity format. The core formulas are: (xy)^n = x^n y^n a^3 + b^3 = (a + b)(a^2 - ab + b^2) Step-by-Step Solution: Let's resolve the numerator first to reveal the true value of our variable a. We know that 8 = 2^3. Rewrite the first term in the numerator: 8(3.75)^3 = 2^3 × (3.75)^3 Using exponent rules, combine the bases: (2 × 3.75)^3 = (7.5)^3 So, the numerator is actually (7.5)^3 + 1^3. This tells us that a = 7.5 and b = 1. Now, let's verify if the denominator matches the a^2 - ab + b^2 pattern required for the shortcut: a^2 - ab + b^2 = (7.5)^2 - (7.5 × 1) + 1^2 = (7.5)^2 - 7.5 + 1 = (7.5)^2 - 6.5 The calculated pattern perfectly matches the given denominator! The entire expression represents: (a^3 + b^3/a^2 - ab + b^2) Which simplifies to (a + b). Calculate the final result: 7.5 + 1 = 8.5 Exam Strategy & Shortcut: Test setters love masking numbers. Whenever you see a coefficient like 8 or 27 next to a cubed term, instantly pull it inside the parentheses as its cube root (2 or 3). Doing 2 × 3.75 = 7.5 immediately reveals the hidden variable a. From there, 7.5 + 1 = 8.5 is a straightforward mental step. Common Pitfall: Attempting to compute 3.75^3 manually or trying to expand 7.5^2. This question is designed specifically to punish brute-force arithmetic. If a calculation looks impossibly tedious, step back—you are missing an algebraic substitution. Final Answer: Therefore, the correct answer is 8.5.

Question 14

$$ \frac{(4.53 - 3.07)^2}{(3.07 - 2.15)(2.15 - 4.53)} + \frac{(3.07 - 2.15)^2}{(2.15 - 4.53)(4.53 - 3.07)} + \frac{(2.15 - 4.53)^2}{(4.53 - 3.07)(3.07 - 2.15)} $$  is simplified to

Accepted Answer

Concept & Formula: This problem leverages a powerful conditional algebraic identity involving cyclic variables. The governing logic is: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. Step-by-Step Solution: To simplify the messy expression, assign variables to each unique bracketed difference: • Let a = 4.53 - 3.07 • Let b = 3.07 - 2.15 • Let c = 2.15 - 4.53 First, test the condition by adding the variables together: a + b + c = (4.53 - 3.07) + (3.07 - 2.15) + (2.15 - 4.53) a + b + c = 4.53 - 4.53 + 3.07 - 3.07 + 2.15 - 2.15 = 0 Because they sum to zero, the rule a^3 + b^3 + c^3 = 3abc is actively unlocked. Next, rewrite the original fraction expression using our variables: (a^2/bc) + (b^2/ca) + (c^2/ab) To add these fractions, find a common denominator, which is abc. Multiply the numerator and denominator of each fraction by the missing variable: (a × a^2/abc) + (b × b^2/abc) + (c × c^2/abc) (a^3 + b^3 + c^3/abc) Now, substitute the active rule (a^3 + b^3 + c^3 = 3abc) into the numerator: (3abc/abc) Cancel out the common term abc from the top and bottom. The final result is simply 3. Exam Strategy & Shortcut: Whenever you see a cyclic pattern of differences—where numbers chain together like (x-y), (y-z), (z-x)—they are guaranteed to sum to zero. If you encounter an expression formatted as squares over products of these cyclic differences (e.g., a^2/bc + b^2/ca + c^2/ab), bypass all algebra. The mathematical structure dictates the answer will always reduce to exactly 3. Common Pitfall: Attempting to perform the manual arithmetic. Calculating (4.53 - 3.07)^2 by hand will consume an enormous amount of time and almost guarantee an arithmetic error. This question is a pure test of algebraic pattern recognition, not calculation speed. Final Answer: Therefore, the correct answer is 3.

Question 15

Solve $\frac{3}{5}$ of $\frac{4}{7}$ of $\frac{5}{12}$ of $1015 = x$

Accepted Answer

Concept & Formula The word "of" in mathematical word problems and simplification questions translates directly to multiplication. The strategy is to multiply all the fractions together first and cancel out common terms in the numerators and denominators before multiplying by the large whole number. Step-by-Step Solution • Write out the full equation replacing "of" with multiplication signs: x = (3/5) × (4/7) × (5/12) × 1015 • Isolate the fraction part to simplify it first: ( (3 × 4 × 5/5 × 7 × 12) ) • Notice that the 5 in the numerator cancels with the 5 in the denominator. • Notice that 3 × 4 = 12 in the numerator, which perfectly cancels out the 12 in the denominator. • The entire fraction expression simplifies to: (1/7) • Now multiply this simplified fraction by the whole number: x = (1/7) × 1015 • Perform the final division: 1015 \div 7 = 145 Exam Strategy & Shortcut Never multiply the first fraction by the large number immediately. Always scan the chain of fractions for cross-cancellation. In this problem, visually spotting that (3 × 4) cancels 12, and 5 cancels 5, reduces the entire problem to a single step: 1015 \div 7, taking mere seconds. Common Pitfall A common time-wasting mistake is trying to solve the chain sequentially from left to right (e.g., finding (3/5) of something, then (4/7) of that result). This creates messy intermediate decimals or large irreducible fractions. Final Answer Therefore, the correct answer is 145.

Question 16

Solve $1\frac{1}{8} + 1\frac{6}{7} + 3\frac{3}{5} = x$

Accepted Answer

Concept & Strategy When adding multiple mixed fractions, the most efficient method is to split the numbers into their integer components and their fractional components. Process them separately before recombining. A(b/c) + X(y/z) = (A + X) + ((b/c) + (y/z)) Step-by-Step Solution • Separate the integer and fraction parts from the given equation: (1 + 1 + 3) + ((1/8) + (6/7) + (3/5)) • Sum the integer parts: 1 + 1 + 3 = 5 • Find the Least Common Multiple (LCM) for the denominators 8, 7, and 5 to add the fractions. Since they share no common factors, multiply them: LCM = 8 × 7 × 5 = 280 • Convert each fraction to have the common denominator of 280: (1 × 35/8 × 35) = (35/280) (6 × 40/7 × 40) = (240/280) (3 × 56/5 × 56) = (168/280) • Add the numerators of the converted fractions together: (35 + 240 + 168/280) = (443/280) • Convert the improper fraction back into a mixed fraction: (443/280) = 1(163/280) • Recombine the integer sum and the new mixed fraction: 5 + 1(163/280) = 6(163/280) Exam Strategy & Shortcut Instead of immediately jumping to find the massive LCM of 280, add the whole numbers first to get 5. Glancing at the fractions (6/7) (which is nearly 1) and (3/5) (more than half), you know their sum will add at least 1 more to your total integer. This means the final integer part must be 6. Looking at the options, (b) is the only choice starting with 6, allowing you to bypass the heavy fraction math entirely. Common Pitfall A frequent error is converting all mixed fractions into improper fractions right at the start (e.g., turning 3(3/5) into (18/5)). This results in very large, unwieldy numerators that must be multiplied for the common denominator, drastically increasing calculation time and the likelihood of arithmetic mistakes. Final Answer Therefore, the correct answer is 6(163/280).

Question 17

Solve $323.46 + 23.04 - 43.17 - x = 303$

Accepted Answer

Concept & Strategy This is a straightforward decimal arithmetic problem. The most effective approach is to follow the order of operations by grouping and solving the additions first, then the subtractions, and finally isolating the unknown variable using basic algebraic principles. Step-by-Step Solution • Group the positive numbers on the left side of the equation and sum them: 323.46 + 23.04 = 346.50 • Substitute this sum back into the equation: 346.50 - 43.17 - x = 303 • Perform the subtraction on the left side: 346.50 - 43.17 = 303.33 • Now the equation is simplified to: 303.33 - x = 303 • Rearrange the equation to solve for x by moving x to the right side and 303 to the left side: 303.33 - 303 = x • Calculate the final difference: x = 0.33 Exam Strategy & Shortcut Use the Unit Digit or "Decimal Digit" method to bypass full calculations. Look only at the hundredths place of the numbers: 0.06 + 0.04 - 0.07 = 0.10 - 0.07 = 0.03. This means the final answer for x must end in a 3. Since the right side of the equation is a whole integer (303.00), the decimal portion of x must be exactly 0.33 to balance the 303.33 on the left. This immediately points to option (a) without doing full vertical math. Common Pitfall A common mistake is misaligning the decimal points when doing vertical addition or subtraction, especially when numbers have different amounts of digits before the decimal. Always ensure the decimal points are perfectly stacked. Final Answer Therefore, the correct answer is 0.33.

Question 18

Solve $1599 \div 39.99 + \frac{4}{5} 	imes 2449 - 120.05 = x$

Accepted Answer

Concept & Strategy This is an approximation problem, signaled by the complex decimals and widely spaced answer choices. Round decimals to their nearest whole numbers to drastically speed up calculation while maintaining sufficient accuracy. Step-by-Step Solution • Approximate the given values to their nearest integers: 1599 \approx 1600 39.99 \approx 40 2449 \approx 2450 120.05 \approx 120 • Rewrite the equation using BODMAS rules (Division and Multiplication before Addition and Subtraction): 1600 \div 40 + (4/5) × 2450 - 120 = x • Perform the division: 1600 \div 40 = 40 • Perform the multiplication: (4/5) × 2450 = 4 × 490 = 1960 • Substitute the results back into the simplified equation: 40 + 1960 - 120 2040 - 120 = 1880 Exam Strategy & Shortcut Recognize the approximation cues instantly. 1599 \div 39.99 is intentionally designed to look ugly but perfectly rounds to 1600 \div 40 = 40. Treat (4/5) × 2449 as (4/5) × 2450. Mentally calculate 2450 \div 5 = 490, then 490 × 4 = 1960. Add 40 to get 2000, and drop the 120. The entire problem can be mapped out mentally in under 15 seconds. Common Pitfall Attempting to calculate exact values (like finding the precise result of (4/5) × 2449 = 1959.2) wastes immense amounts of time and creates messy decimals that often lead to addition/subtraction errors in the final steps. Never do exact math on an approximation question. Final Answer Therefore, the correct answer is 1880.

Question 19

Solve $\frac{294 \div 14 	imes 5 + 11}{x} = 8^2 \div 5 + 1.7$

Accepted Answer

Concept & Logic: This problem requires strict adherence to the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). The equation must be simplified by evaluating the numerator on the left and the entire expression on the right before isolating x. Step-by-Step Solution: • Given Equation: (294 \div 14 × 5 + 11/x) = 8^2 \div 5 + 1.7 • Step 1: Simplify the Left-Hand Side (LHS) Numerator According to BODMAS, perform division first, then multiplication, then addition. 294 \div 14 = 21 21 × 5 = 105 105 + 11 = 116 So, the LHS becomes (116/x). • Step 2: Simplify the Right-Hand Side (RHS) Resolve the exponent (order) first, then division, then addition. 8^2 = 64 64 \div 5 = 12.8 12.8 + 1.7 = 14.5 So, the RHS becomes 14.5. • Step 3: Solve for x Equate the simplified sides: (116/x) = 14.5 x = (116/14.5) To remove the decimal, multiply the numerator and denominator by 10: x = (1160/145) Dividing gives: x = 8 Exam Strategy & Shortcut: Simplify expressions in chunks mentally. For LHS: 294/14 = 21; 21 × 5 = 105; 105+11 = 116. For RHS: 64/5 = 12.8; 12.8+1.7 = 14.5. Now you have 116 = 14.5 × x. Instead of long division, check the unit digits or use multiples. 14.5 × 2 = 29. 29 × 4 = 116. Therefore, x = 2 × 4 = 8. Common Pitfall: Failing to follow BODMAS in the numerator is the most common mistake (e.g., multiplying before dividing or adding before multiplying). Also, arithmetic errors frequently happen when dividing by decimals like 14.5. Final Answer: Therefore, the correct answer is 8.

Question 20

Solve $2.5 	imes 4.8 + 7.2 	imes 1.5 - 1.2 	imes 14 = x$

Accepted Answer

Concept & Logic: The problem requires applying the standard order of operations (BODMAS/PEMDAS) to decimal arithmetic. Multiplication must be performed before addition and subtraction. Step-by-Step Solution: • Step 1: Isolate and perform the multiplication operations first. • 2.5 × 4.8 = 12 • 7.2 × 1.5 = 10.8 • 1.2 × 14 = 16.8 • Step 2: Substitute these products back into the original equation. • Equation becomes: 12 + 10.8 - 16.8 • Step 3: Perform addition and subtraction from left to right. • 22.8 - 16.8 = 6 Exam Strategy & Shortcut: To multiply decimals quickly, temporarily ignore the decimal points, perform integer multiplication, and then place the decimal back based on the total decimal places. For example, for 2.5 × 4.8, calculate 25 × 48. An easy way to do 25 × 48 is 100/4 × 48 = 1200. Since there are two decimal places in total (2.5 and 4.8), shift the decimal two spots to the left: 12.00. Common Pitfall: Students often make alignment errors when adding or subtracting the final terms, such as subtracting 16.8 from 10.8 incorrectly or misplacing the decimal point during the initial multiplication steps. Final Answer: Therefore, the correct answer is 6.

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