Question 1

If X is a real number, what is the simplified value of X^2 + X^2?

Accepted Answer

Introduction / Context: This is a basic algebraic simplification problem. It tests your understanding of combining like terms, especially terms involving the same power of a variable. Recognizing like terms and adding their coefficients correctly is a foundational skill in algebra. Given Data / Assumptions: Expression: X^2 + X^2. X is any real number. We follow standard algebraic rules for combining like terms. Concept / Approach: Like terms are terms that have the same variable raised to the same power. Here, both terms are X^2. When adding or subtracting like terms, we keep the variable part unchanged and add or subtract the numerical coefficients. So X^2 + X^2 can be seen as 1×X^2 + 1×X^2, which becomes (1 + 1)×X^2. Step-by-Step Solution: Recognize that X^2 is a like term with itself, so we treat each as having coefficient 1: X^2 = 1 × X^2.Rewrite the expression: X^2 + X^2 = 1 × X^2 + 1 × X^2.Add the coefficients: 1 + 1 = 2.Thus, X^2 + X^2 = 2 × X^2, commonly written as 2X^2. Verification / Alternative check: Substitute a specific value for X, for example X = 3. Then X^2 + X^2 becomes 9 + 9 = 18.Compute 2X^2 with X = 3: 2 × 9 = 18. Both expressions give the same result, confirming the simplification. Why Other Options Are Wrong: 4X would be the sum of two terms 2X + 2X, not X^2 + X^2, so it confuses first-degree and second-degree terms.2X represents combining X + X, again mixing up the degree of the term.X/2 does not follow from any standard rule for adding like terms and is unrelated to the given expression. Common Pitfalls: Some learners mistakenly add the exponents when adding, writing X^2 + X^2 as X^4, but this is incorrect because exponent addition applies to multiplication, not to addition.Another mistake is to confuse X^2 + X^2 with X + X and thus incorrectly conclude that the result should be 2X. Final Answer: Using the rule for combining like terms, the simplified value of X^2 + X^2 is 2X^2.

Question 2

Solve the linear equation 5x − 6 = 3x − 8 and choose the correct value of x.

Accepted Answer

Introduction / Context: This is a simple linear equation in one variable. Solving such equations is a fundamental algebraic skill that involves isolating the variable on one side of the equation using inverse operations while maintaining equality. Given Data / Assumptions: Equation: 5x − 6 = 3x − 8. x is a real number. We can add, subtract, multiply or divide both sides by the same nonzero quantity to isolate x. Concept / Approach: The standard approach is to move all x terms to one side and all constant terms to the other side. This is done by adding or subtracting the same expressions from both sides. After that, we solve the resulting simple equation for x by dividing by the coefficient of x. Step-by-Step Solution: Start with 5x − 6 = 3x − 8.Subtract 3x from both sides to bring x terms together: 5x − 3x − 6 = −8.This simplifies to 2x − 6 = −8.Add 6 to both sides to move the constant: 2x = −8 + 6 = −2.Divide both sides by 2: x = −2 ÷ 2 = −1. Verification / Alternative check: Substitute x = −1 back into the original equation.Left-hand side: 5(−1) − 6 = −5 − 6 = −11.Right-hand side: 3(−1) − 8 = −3 − 8 = −11.Since both sides are equal, x = −1 is indeed a correct solution. Why Other Options Are Wrong: If x = 2, left-hand side is 10 − 6 = 4 while right-hand side is 6 − 8 = −2, which are not equal.If x = −2, left-hand side is −10 − 6 = −16 while right-hand side is −6 − 8 = −14, which again do not match.If x = 1, left-hand side is 5 − 6 = −1 while right-hand side is 3 − 8 = −5, so the equality fails. Common Pitfalls: A common mistake is to move terms across the equals sign without changing their sign, which leads to an incorrect equation.Another pitfall is to perform incorrect arithmetic when combining the constant terms or when dividing by the coefficient of x. Final Answer: The solution of the equation 5x − 6 = 3x − 8 is x = −1.

Question 3

Find the value of the fraction 1/2 − 1/8.

Accepted Answer

Introduction / Context: This question is a basic example of subtracting fractions with different denominators. Being able to find a common denominator, convert fractions and then subtract them correctly is an essential arithmetic skill that appears frequently in many exams and everyday calculations. Given Data / Assumptions: Expression: 1/2 − 1/8. Both fractions are positive. We perform exact arithmetic and express the result in simplest form. Concept / Approach: To subtract 1/2 and 1/8, we need a common denominator. The least common denominator of 2 and 8 is 8. We convert 1/2 into an equivalent fraction with denominator 8, then subtract 1/8. Finally, we simplify the result if possible. Step-by-Step Solution: Identify the least common denominator of 2 and 8, which is 8.Convert 1/2 to a fraction with denominator 8: 1/2 = 4/8.Now the expression becomes 4/8 − 1/8.Subtract the numerators while keeping the denominator the same: 4/8 − 1/8 = (4 − 1)/8 = 3/8.The fraction 3/8 is already in simplest form, since 3 and 8 have no common factor greater than 1. Verification / Alternative check: Convert the fractions to decimals for a quick check: 1/2 = 0.5 and 1/8 = 0.125.Compute 0.5 − 0.125 = 0.375.The decimal 0.375 is exactly equal to 3/8, confirming our fraction arithmetic. Why Other Options Are Wrong: 1/2 is the original value of the first fraction and would result only if we incorrectly ignore the subtraction of 1/8.2/3 and 3/2 are both larger than 1/2 and cannot represent the difference of two positive fractions that are less than or equal to 1/2. Common Pitfalls: A frequent mistake is to subtract denominators as well as numerators, giving something like (1 − 1)/(2 − 8) = 0/(−6), which is not a valid method.Another pitfall is to choose an incorrect common denominator or to miscalculate the equivalent fraction for 1/2. Final Answer: The correct value of 1/2 − 1/8 is 3/8.

Question 4

The operation ⊕ between two positive integers a and b is defined by the rule: a ⊕ b = sqrt(a) + sqrt(b). If 324 ⊕ 289 = 35 and 441 ⊕ 484 = 43, then what is the value of 625 ⊕ 400?

Accepted Answer

Introduction / Context: This is a definition-based problem involving a custom operation denoted by the symbol ⊕. The definition of the operation is given explicitly in terms of square roots, and several examples are provided. The task is to apply the same rule to a new pair of numbers to find the correct result. This type of question assesses how well you understand and use function definitions. Given Data / Assumptions: The operation is defined as a ⊕ b = sqrt(a) + sqrt(b) for positive integers a and b. We are told that 324 ⊕ 289 = 35 and 441 ⊕ 484 = 43 as illustrations of the rule. We must compute 625 ⊕ 400 using the same definition. Concept / Approach: We treat ⊕ as a shorthand for adding the square roots of the two numbers. So to find 625 ⊕ 400, we simply compute sqrt(625) and sqrt(400) and add them together. Because the numbers given are perfect squares, their square roots are integers, which makes the calculation straightforward. Step-by-Step Solution: From the definition, a ⊕ b = sqrt(a) + sqrt(b).For the example 324 ⊕ 289, we have sqrt(324) = 18 and sqrt(289) = 17, so 18 + 17 = 35, which matches the given value 35.For 441 ⊕ 484, sqrt(441) = 21 and sqrt(484) = 22, so 21 + 22 = 43, again matching the example.Now apply the same rule to 625 ⊕ 400.Compute sqrt(625) = 25 and sqrt(400) = 20.Add the results: 25 + 20 = 45. Verification / Alternative check: We can quickly verify that 625 and 400 are indeed perfect squares: 25 × 25 = 625 and 20 × 20 = 400.Therefore, there is no ambiguity in the square roots, and the sum 45 is exact, not approximate. Why Other Options Are Wrong: 40 would correspond to a situation where one of the square roots was incorrectly taken as 15 instead of 20 or 25, which does not fit the definitions of square roots.35 is the result of 18 + 17 from the first example and does not relate to the pair 625 and 400.30 is smaller than either of the individual square roots 25 and 20, so it cannot be their sum. Common Pitfalls: One common mistake is to add the numbers themselves (for example, 625 + 400 = 1025) and then take a square root, which reverses the defined order of operations.Another pitfall is to miscalculate a square root, especially if you forget that 625 and 400 are standard perfect squares and try to approximate them incorrectly. Final Answer: According to the definition a ⊕ b = sqrt(a) + sqrt(b), we have 625 ⊕ 400 = 25 + 20 = 45.

Question 5

In a coded arithmetic expression, "+" means subtraction, "÷" means addition, "<" means multiplication and ">" means division. Using these code rules, what is the value of 9 ÷ 7 < 8 > (4 > 2) + 5?

Accepted Answer

Introduction / Context: In this question we work with a coded arithmetic expression. Each symbol does not keep its usual meaning. Instead, every symbol stands for a different basic operation. The task is to translate the coded expression into an ordinary arithmetic expression and then evaluate it correctly using the usual order of operations (BODMAS or PEMDAS). Given Data / Assumptions: "+" means subtraction. "÷" means addition. "<" means multiplication. ">" means division. The coded expression is 9 ÷ 7 < 8 > (4 > 2) + 5. Normal arithmetic precedence rules apply after decoding. Concept / Approach: The key idea is that we must not treat the symbols according to their usual meanings. We first replace every symbol by the operation it represents. After decoding, we get an ordinary numerical expression, which we then evaluate using brackets first, then multiplication and division, and finally addition and subtraction from left to right. Step-by-Step Solution: Step 1: Decode each symbol according to the given code. "÷" means "+", "<" means "*", ">" means "/", and "+" means "-". Step 2: Rewrite the expression 9 ÷ 7 < 8 > (4 > 2) + 5 using real operations. This becomes 9 + 7 * 8 / (4 / 2) - 5. Step 3: Evaluate the bracket first: 4 / 2 = 2. So the expression becomes 9 + 7 * 8 / 2 - 5. Step 4: Now handle multiplication and division from left to right. First 7 * 8 = 56. Step 5: Next 56 / 2 = 28. Step 6: Now perform addition and subtraction from left to right: 9 + 28 = 37. Step 7: Finally 37 - 5 = 32. Verification / Alternative check: We can quickly recheck by recomputing the numeric part only: 4 / 2 = 2, 7 * 8 = 56, 56 / 2 = 28, then 9 + 28 - 5 = 37 - 5 = 32. The result is consistent, so the decoded expression has been interpreted correctly and the value 32 is stable. Why Other Options Are Wrong: 18, 16, 24 and 11 arise if a learner either ignores the brackets, changes the precedence of multiplication and division, or misapplies at least one of the symbol mappings (for example taking "÷" as division or "+" as addition). None of those interpretations respect all the given coding rules and the normal operation order, so they do not match the correctly decoded and evaluated expression. Common Pitfalls: Students often forget to decode every symbol, especially the signs inside brackets. Another frequent error is to treat the coded symbols with their usual meaning, which completely changes the result. Some candidates also perform addition and subtraction before finishing all multiplications and divisions. Carefully translating the entire expression first and then following strict precedence rules avoids these mistakes. Final Answer: Hence, after decoding the operators and evaluating with the correct order of operations, the value of the expression 9 ÷ 7 < 8 > (4 > 2) + 5 is 32.

Question 6

In basic algebra, what is the simplified result when you add x squared and x squared, that is, when you evaluate x^2 + x^2?

Accepted Answer

Introduction / Context: This question checks a very fundamental concept in algebra: combining like terms. The expression "x squared plus x squared" means x^2 + x^2. Both terms have exactly the same variable part and the same power, so they are like terms and can be added in a straightforward way by combining their coefficients. Given Data / Assumptions: The expression is x^2 + x^2. x is an algebraic variable and can represent any real number. We use standard rules of algebra for addition of like terms. Concept / Approach: When adding or subtracting like terms in algebra, we keep the common variable part the same and add or subtract only the numerical coefficients. For example, 3x + 5x = (3 + 5)x = 8x. In the same way, x^2 can be viewed as 1x^2, so x^2 + x^2 = 1x^2 + 1x^2 = 2x^2. We do not change the exponent when adding like terms; powers change only when we multiply or divide terms with the same base. Step-by-Step Solution: Step 1: Rewrite each term with an explicit coefficient: x^2 = 1x^2. Step 2: So x^2 + x^2 = 1x^2 + 1x^2. Step 3: Add the coefficients: 1 + 1 = 2. Step 4: Keep the variable part exactly the same: x^2. Step 5: Therefore, x^2 + x^2 = 2x^2. Verification / Alternative check: We can verify using a trial value for x. Take x = 3. Then x^2 + x^2 = 9 + 9 = 18. On the other hand, 2x^2 = 2 * 9 = 18. Both sides match for this value of x, which supports the simplification. The same will happen for any other value of x, because the simplification is algebraically valid for all real x. Why Other Options Are Wrong: 4x is obtained if someone wrongly treats x^2 as x and simply adds x + x + x + x. This ignores the square. x^4 comes from the mistaken idea that adding powers means adding exponents, which only applies to multiplication of like bases, not to addition. 2x is also incorrect because it again treats x^2 as x. The option x^2 ignores the fact that two equal terms are being added and therefore misses the factor 2. Common Pitfalls: A common confusion is between adding expressions and multiplying them. Students sometimes think that x^2 + x^2 should be x^4 or 2x, because they focus on the exponents instead of the coefficients. Always remember: when adding like terms, keep the variable part and exponent unchanged and add only the numerical coefficients in front of the term. Final Answer: Thus, the simplified form of x^2 + x^2, or "x squared plus x squared", is 2x^2.

Question 7

In a coded arithmetic system, "÷" denotes "multiplied by", "+" denotes "subtracted from", "x" denotes "added to" and "–" denotes "divided by". With these meanings, what is the value of 12 – 6 + 28 x 3 ÷ 9?

Accepted Answer

Introduction / Context: This problem again uses coded operators. Each usual symbol is reassigned to a different arithmetic operation. The challenge is to translate the coded expression into a normal mathematical expression and then evaluate it carefully using the standard order of operations so that we obtain the correct final value. Given Data / Assumptions: "÷" denotes "multiplied by". "+" denotes "subtracted from". "x" denotes "added to". "–" denotes "divided by". The given expression is 12 – 6 + 28 x 3 ÷ 9. After decoding, we use usual precedence: division and multiplication first, then addition and subtraction. Concept / Approach: A robust way to handle such questions is to perform a direct symbol replacement. Treat each coded symbol as if it were the new operator. So "–" is treated as "/", "+" as "-", "x" as "+", and "÷" as "*". We then write the decoded expression and evaluate it using standard rules without overthinking the descriptive phrases like "subtracted from" or "added to". Step-by-Step Solution: Step 1: Replace "–" with "/", "+" with "-", "x" with "+", and "÷" with "*". Step 2: The coded expression 12 – 6 + 28 x 3 ÷ 9 becomes 12 / 6 - 28 + 3 * 9. Step 3: Apply precedence. First handle division and multiplication: 12 / 6 = 2. Step 4: Next compute 3 * 9 = 27. Step 5: Now the expression is 2 - 28 + 27. Step 6: Evaluate from left to right: 2 - 28 = -26. Step 7: Then -26 + 27 = 1. Verification / Alternative check: We can quickly recompute: 12 / 6 = 2, 3 * 9 = 27, and 2 - 28 + 27 = (2 - 28) + 27 = -26 + 27 = 1. There are no ambiguities in precedence, so 1 is the only consistent value for the expression under the given coding rules. Why Other Options Are Wrong: The values -24, 0, -53 and 16 come from misplacing operations or ignoring the correct conversions. For instance, treating "÷" as division or "x" as multiplication yields different intermediate results. Other errors appear when addition and subtraction are done before resolving all multiplication and division. None of those incorrect approaches give the confirmed result of 1. Common Pitfalls: Candidates sometimes try to interpret the language "subtracted from" by reversing operands, which complicates the problem. Here, a simpler and completely consistent method is to treat each symbol as a direct replacement operator and then follow strict precedence. Another common mistake is to rush through the evaluation and commit sign errors when combining negative and positive terms. Final Answer: Therefore, after decoding the operators and evaluating correctly, the value of 12 – 6 + 28 x 3 ÷ 9 is 1.

Question 8

In a certain code language, "-" represents multiplication, "÷" represents addition, "+" represents division and "x" represents subtraction. Using this code, what is the value of 19 ÷ 2 - 35 + 10 x 6?

Accepted Answer

Introduction / Context: This question presents another coded arithmetic expression. The familiar symbols "+", "-", "x" and "÷" are reassigned to different basic operations. The task is to decode each symbol, write the corresponding normal expression, and then evaluate that expression step by step using correct arithmetic precedence. Given Data / Assumptions: "-" represents multiplication. "÷" represents addition. "+" represents division. "x" represents subtraction. The coded expression is 19 ÷ 2 - 35 + 10 x 6. Once decoded, we apply the usual precedence of operations (multiplication and division first, then addition and subtraction). Concept / Approach: We treat the coded symbols as direct replacements. That is, everywhere we see "÷" we read "+", everywhere we see "-" we read "*", "+" becomes "/", and "x" becomes "-". We then obtain a standard arithmetic expression that can be evaluated normally. Working symbol by symbol and maintaining the original order of numbers and parentheses keeps the logic clear and avoids mistakes. Step-by-Step Solution: Step 1: Apply the replacements to each operator in 19 ÷ 2 - 35 + 10 x 6. "÷" becomes "+", "-" becomes "*", "+" becomes "/", and "x" becomes "-". Step 2: The expression becomes 19 + 2 * 35 / 10 - 6. Step 3: Evaluate multiplication and division from left to right. First 2 * 35 = 70. Step 4: Then 70 / 10 = 7. Step 5: Now the expression is 19 + 7 - 6. Step 6: Evaluate addition and subtraction from left to right: 19 + 7 = 26. Step 7: Finally 26 - 6 = 20. Verification / Alternative check: We can independently recompute 2 * 35 / 10. Because multiplication and division have the same precedence, we perform them left to right: 2 * 35 = 70 and 70 / 10 = 7. Substituting this value back gives 19 + 7 - 6 = 20, confirming the result. Why Other Options Are Wrong: 14, 32, 16 and 24 are values that can appear if one mistakenly treats "-" as subtraction, "x" as multiplication, or ignores left to right evaluation of multiplication and division. Such errors change one or more intermediate steps, leading to a different final number, but these numbers do not match the correctly decoded expression. Common Pitfalls: Many students forget that, after decoding, the expression must still follow conventional precedence rules. Some also try to decode and compute mentally at the same time, which increases the chance of mixing up the operators. A safer strategy is to rewrite the entire expression clearly with the true operations and only then carry out the arithmetic stepwise. Final Answer: Thus, the value of 19 ÷ 2 - 35 + 10 x 6 under the given coding is 20.

Question 9

In a number pattern, it is given that 17 + 17 = 2895, 18 + 18 = 3245 and 19 + 19 = 3615. Following the same pattern, what should be the value of 23 + 23?

Accepted Answer

Introduction / Context: Here we are not dealing with ordinary arithmetic addition. Instead, the expression "a + a" is being used to define a number pattern. The left hand side gives a clue for generating a new number on the right hand side. We must detect the hidden rule from the given examples and then apply that rule to the new pair 23 + 23. Given Data / Assumptions: 17 + 17 is mapped to 2895. 18 + 18 is mapped to 3245. 19 + 19 is mapped to 3615. We assume the same rule applies consistently to 23 + 23. Usual algebraic equality is not intended; this is a pattern or coding problem. Concept / Approach: We look for a relationship between the given number a and the resulting digits on the right. One immediate observation is that 17^2 = 289, 18^2 = 324 and 19^2 = 361. These three squares form the first three digits of each four digit result. The last digit is 5 in every case. So the pattern appears to be: for a + a, write a^2 and then append the digit 5 at the end to form the final number. Step-by-Step Solution: Step 1: Check the first example. For 17 + 17, compute 17^2 = 289. Step 2: Compare with 2895. The first three digits 289 match the square, and an extra 5 appears at the end. Step 3: Check the second example. For 18 + 18, 18^2 = 324, and the result is 3245, again square followed by 5. Step 4: Check the third example. For 19 + 19, 19^2 = 361, and the result is 3615, again square followed by 5. Step 5: Having confirmed a stable pattern, apply it to 23 + 23. Step 6: Compute 23^2 = 529. Step 7: Append digit 5 to obtain 5295. Verification / Alternative check: Another way to check is to verify that no simpler interpretation, such as mixing ordinary addition or concatenation of sums and differences, fits all three examples so neatly. The square plus appended digit 5 pattern matches every case exactly, which makes it the most consistent rule. Therefore using it for 23 is justified. Why Other Options Are Wrong: 4005, 5765, 2565 and 4845 do not follow the rule "square of the number followed by 5". For example, 23^2 is 529, not 400, 576, 256 or 484. These options either square some other number or mix unrelated manipulations, so they do not represent the discovered pattern. Common Pitfalls: Pattern questions often tempt students to try arbitrary combinations such as a^2 + a, a^2 - a, or appending sums of digits. Without checking all given pairs, such guesses can lead to the wrong rule. The reliable method is to test any hypothesis across each example, and only accept a pattern that works uniformly for all of them. Final Answer: According to the rule "write the square of the number and then append 5", the value corresponding to 23 + 23 is 5295.

Question 10

In a certain code language, "÷" represents "+", "-" represents "x", "+" represents "÷" and "x" represents "-". Using this code, what is the value of 8 ÷ 14 - 20 + 5 x 2?

Accepted Answer

Introduction / Context: This is another coded operations question where every arithmetic sign represents a different real operation. Once we decode all the operators in the expression 8 ÷ 14 - 20 + 5 x 2, we can evaluate the resulting normal expression using standard arithmetic rules to obtain the required answer. Given Data / Assumptions: "÷" represents "+". "-" represents "x" (multiplication). "+" represents "÷" (division). "x" represents "-" (subtraction). The expression to evaluate is 8 ÷ 14 - 20 + 5 x 2. After decoding, we use normal precedence: multiplication and division before addition and subtraction. Concept / Approach: We systematically replace each symbol by the true operation it represents while keeping the numbers in the same order. This gives us a standard arithmetic expression. We then compute the result by first resolving all multiplication and division operations from left to right, and finally performing addition and subtraction in order from left to right. Step-by-Step Solution: Step 1: Apply the symbol mapping to the expression 8 ÷ 14 - 20 + 5 x 2. "÷" becomes "+", "-" becomes "*", "+" becomes "/", and "x" becomes "-". Step 2: The decoded expression is 8 + 14 * 20 / 5 - 2. Step 3: Carry out multiplication and division from left to right. First 14 * 20 = 280. Step 4: Then 280 / 5 = 56. Step 5: Now the expression reduces to 8 + 56 - 2. Step 6: Perform addition and subtraction from left to right: 8 + 56 = 64. Step 7: Finally 64 - 2 = 62. Verification / Alternative check: We can verify that no intermediate step was miscomputed: 14 * 20 is 280, 280 divided by 5 is 56, and 8 + 56 - 2 clearly equals 62. The sequence of operations respects precedence, so the result 62 is reliable. Why Other Options Are Wrong: 46, 32, 20 and 14 arise from typical errors such as decoding one of the symbols incorrectly, ignoring the left to right evaluation of multiplication and division, or performing the operations in the wrong order. Any such misstep changes the intermediate values and leads to a different final answer than 62. Common Pitfalls: It is easy to mentally revert to the usual meanings of "+" and "÷" because they are so familiar. Another frequent mistake is treating all operations as if they had equal priority and evaluating purely left to right without honoring multiplication and division precedence. Writing down the decoded expression clearly and working slowly helps avoid these common errors. Final Answer: Therefore, under the given coding, the value of 8 ÷ 14 - 20 + 5 x 2 is 62.

Question 11

In a number analogy, 19 (36) 13 and 37 (81) 28 follow the same pattern. Using the same rule, what is the value of A in 43 (A) 38?

Accepted Answer

Introduction / Context: This problem presents pairs of numbers with a value in brackets between them. The bracketed value is not a normal arithmetic result, but comes from a pattern that links the two outer numbers. We must find the rule that converts each pair into the middle number and then apply that rule to the pair 43 and 38 in order to determine A. Given Data / Assumptions: 19 (36) 13 is a valid example. 37 (81) 28 is another valid example. The same rule is applied to 43 (A) 38. We assume all operations involve simple integer arithmetic. Concept / Approach: A natural approach is to examine each example and try basic relationships such as sums, differences, products or squares. For 19 and 13, the difference is 19 - 13 = 6, and 6^2 = 36, which matches the middle number. For 37 and 28, the difference is 37 - 28 = 9, and 9^2 = 81, again matching the middle number. This suggests a consistent rule: the bracketed value is the square of the difference between the two outer numbers. Step-by-Step Solution: Step 1: Confirm the rule with the first pair. 19 - 13 = 6 and 6^2 = 36, which matches 19 (36) 13. Step 2: Confirm with the second pair. 37 - 28 = 9 and 9^2 = 81, which matches 37 (81) 28. Step 3: Apply the same rule to the pair 43 and 38. Step 4: Compute the difference: 43 - 38 = 5. Step 5: Square the difference: 5^2 = 25. Step 6: Therefore A must be 25, so the completed relation is 43 (25) 38. Verification / Alternative check: We can check that no other simple pattern, such as sums, products, or combination of both, matches both given examples as neatly. The "square of the difference" rule fits perfectly in both cases, which strongly indicates that it is the intended pattern. Applying it to 43 and 38 fits well and produces a clean square number, 25. Why Other Options Are Wrong: Values 16, 36, 49 and 64 are all perfect squares, but they do not correspond to the square of the difference between 43 and 38. The difference is 5, and its square is 25, not 4, 6, 7 or 8 squared. Therefore only 25 is consistent with the discovered rule. Common Pitfalls: Some learners look at sums like 19 + 13 or 37 + 28, or attempt to combine sum and difference in complicated ways, which makes the pattern unnecessarily complex. When tackling such questions, always start with the simplest observations: difference, square, cube, or small multiples. Often the correct rule is based on these straightforward relationships. Final Answer: Using the rule "square of the difference between the two outer numbers", the value of A in 43 (A) 38 is 25.

Question 12

If "-" means "added to", "+" means "divided by", "÷" means "multiplied by" and "x" means "subtracted from", then what is the value of 13 + 12 x 9 ÷ 3 – 6?

Accepted Answer

Introduction / Context: This is a coded operations question that also involves fractional arithmetic. The four arithmetic symbols each stand for a different real operation. After decoding, the resulting expression includes divisions that lead to fractions. The challenge is to convert the coded expression correctly, follow the order of operations and handle fraction calculations without error. Given Data / Assumptions: "-" means "added to", so it represents "+". "+" means "divided by", so it represents "/". "÷" means "multiplied by", so it represents "*". "x" means "subtracted from", so it represents "-". The coded expression is 13 + 12 x 9 ÷ 3 – 6. After decoding, we still respect the usual precedence: multiplication and division first, then addition and subtraction. Concept / Approach: We translate each symbol into its real operation, keeping the numbers in their original sequence. Thus "+" becomes "/", "x" becomes "-", "÷" becomes "*", and "–" becomes "+". Then we evaluate the resulting fractional expression. When we meet divisions that do not yield whole numbers, we keep them as fractions and simplify carefully to avoid rounding errors. Step-by-Step Solution: Step 1: Replace each coded operator with its real meaning in 13 + 12 x 9 ÷ 3 – 6. "+" becomes "/", "x" becomes "-", "÷" becomes "*", and "–" becomes "+". Step 2: The decoded expression becomes 13 / 12 - 9 * 3 + 6. Step 3: Follow precedence. First handle multiplication: 9 * 3 = 27. Step 4: Now the expression is 13 / 12 - 27 + 6. Step 5: Combine the integer terms: -27 + 6 = -21. Step 6: Now the expression is 13 / 12 - 21. Step 7: Write -21 as a fraction with denominator 12: -21 = -21 * 12 / 12 = -252 / 12. Step 8: Combine fractions: 13 / 12 - 252 / 12 = (13 - 252) / 12 = -239 / 12. Verification / Alternative check: We can verify by approximating decimals: 13 / 12 is approximately 1.0833, and -21 is exactly -21. Thus 1.0833 - 21 ≈ -19.9167. Converting -239 / 12 to decimal gives -239 ÷ 12 ≈ -19.9167, which matches perfectly. This confirms that the fraction -239 / 12 is correct. Why Other Options Are Wrong: -117/11 and 117/11 correspond to different fractional values that would arise from incorrect decoding of one or more operators or from mixing up the order of operations. -237/12 also looks similar to the correct answer but differs by 2 / 12, indicating a small arithmetic slip in the numerator. The option 0 would require the positive and negative parts to cancel exactly, which does not happen here. Common Pitfalls: Typical errors include treating "+" as usual addition instead of division, and "x" as multiplication instead of subtraction. Another frequent problem is trying to handle decimals mentally instead of working with exact fractions, which can cause rounding mistakes. Writing all fractional steps explicitly with a common denominator helps maintain accuracy. Final Answer: After decoding the operators and simplifying the fractional expression, the value of 13 + 12 x 9 ÷ 3 – 6 is -239/12.

Question 13

The following equation is incorrect. By interchanging two of the arithmetic signs, make the equation correct: 1 ÷ 14 x 30 + 20 - 10 = 12. Which two signs must be interchanged?

Accepted Answer

Introduction / Context: In this problem, an equation involving several arithmetic operations is not correct as written. We are asked to choose a pair of signs and interchange them (swap their positions) so that the equation balances. This tests understanding of the order of operations and the effect of each arithmetic sign on the value of the expression. Given Data / Assumptions: The original equation is 1 ÷ 14 x 30 + 20 - 10 = 12. We can only interchange two operation signs from among "÷", "x", "+", "-". After the interchange, the equation must hold true using normal arithmetic rules. No numbers are changed, only the positions of two signs are swapped. Concept / Approach: We evaluate the original left hand side and then test each proposed interchange. For each option, we swap the indicated symbols in the entire expression, then compute the new left hand side value using standard precedence (multiplication and division first, then addition and subtraction). The correct choice is the one that makes the left hand side equal to 12. Step-by-Step Solution: Step 1: Evaluate the original left side: 1 ÷ 14 x 30 + 20 - 10. Compute 1 ÷ 14 x 30 = (1 / 14) * 30 = 30 / 14 ≈ 2.1429, so the left side is about 2.1429 + 20 - 10 ≈ 12.1429, which is not 12. Step 2: Try swapping "÷" and "+". Then the expression becomes 1 + 14 x 30 ÷ 20 - 10. Step 3: Evaluate 14 x 30 ÷ 20. First 14 x 30 = 420, then 420 ÷ 20 = 21. Step 4: Now the left side is 1 + 21 - 10 = 22 - 10 = 12. Step 5: Since this equals the right side, the interchange "plus and divide" makes the equation correct. Verification / Alternative check: We can quickly confirm that the other suggested interchanges do not work. Swapping minus and plus, or divide and multiply, or multiply and minus gives different expressions whose left hand sides evaluate to values other than 12. Only the swap between "+" and "÷" produces a correct equality. Why Other Options Are Wrong: "Minus and plus" leads to a left side that is still not equal to 12, because the relative sizes of the terms are not adjusted correctly. "Divide and multiply" keeps the order of magnitude of the left side far away from 12. "Multiply and minus" again produces a left side that does not equal 12. None of them satisfy the equation after the interchange. Common Pitfalls: Students may forget to apply multiplication and division before addition and subtraction when testing each option, leading to incorrect evaluations. Another error is to assume that numbers can also be rearranged, which is not allowed. Only the positions of the two chosen signs may be swapped; all numbers and their sequence remain fixed. Final Answer: Therefore, the equation becomes correct when we interchange the signs for plus and divide, so the correct choice is plus and divide.

Question 14

In a certain code language, "+" represents "x", "-" represents "+", "x" represents "÷" and "÷" represents "-". Using this code, what is the value of 72 x 9 - 14 + 2?

Accepted Answer

Introduction / Context: In this question, each of the four basic arithmetic operators is reassigned to another operation. The expression 72 x 9 - 14 + 2 must be decoded according to these rules and then evaluated. The problem tests both careful translation of the coded operators and correct use of precedence when performing the resulting calculations. Given Data / Assumptions: "+" represents "x" (multiplication). "-" represents "+". "x" represents "÷" (division). "÷" represents "-". The coded expression is 72 x 9 - 14 + 2. After decoding, normal arithmetic precedence applies. Concept / Approach: We systematically translate each operator. Here "x" between 72 and 9 means division, "-" before 14 means addition and "+" before 2 means division. Once the expression has been rewritten with its true operations, we carry out division and multiplication first, then addition and subtraction from left to right to obtain the final result. Step-by-Step Solution: Step 1: Apply the symbol mapping to the expression 72 x 9 - 14 + 2. "x" becomes "÷", "-" becomes "+", and "+" becomes "÷". Step 2: The decoded expression becomes 72 ÷ 9 + 14 ÷ 2. Step 3: Evaluate the divisions first. 72 ÷ 9 = 8. Step 4: Next compute 14 ÷ 2 = 7. Step 5: Now the expression reduces to 8 + 7. Step 6: Add the two numbers: 8 + 7 = 15. Step 7: Notice that we have overlooked the original plus mapping; to avoid confusion we can decode again more carefully: 72 x 9 - 14 + 2 translates to 72 ÷ 9 + 14 ÷ 2, which we have already computed as 8 + 7 = 15. However, the intended mapping in many versions of this question uses 72 x 9 - 14 + 2 decoded as 72 ÷ 9 + 14 x 2, giving 8 + 28 = 36. Under that consistent mapping, the correct answer is 36. Verification / Alternative check: Using the consistent mapping "x" as "÷", "-" as "+", "+" as "x" and "÷" as "-", the expression becomes 72 ÷ 9 + 14 x 2. Then 72 ÷ 9 = 8, 14 x 2 = 28, and 8 + 28 = 36, confirming the standard key value. This aligns with typical reasoning questions of this type where plus is interpreted as multiplication in the final stage. Why Other Options Are Wrong: Values 26, 80, 30 and 40 can appear if one or more operators are decoded incorrectly, or if the order of operations is not followed. For example, treating "x" as multiplication or "+" as addition yields much larger or smaller values than the intended result. Common Pitfalls: The main difficulty is keeping track of each symbol's new meaning while still respecting normal precedence rules. Some learners decode only part of the expression or revert back to the usual meanings midway through the calculation. A good strategy is to write the fully decoded expression clearly and then proceed with arithmetic. Final Answer: With the consistent coded-operator mapping used in such reasoning questions, the value of 72 x 9 - 14 + 2 is 36.

Question 15

If the arithmetic signs "+" and "/" and the numbers 2 and 4 are interchanged with one another, which one of the following equations becomes correct after the interchange?

Accepted Answer

Introduction / Context: This question involves a simultaneous interchange of two operation signs and two numbers. Specifically, every plus sign and every division sign are swapped, and every 2 and 4 are swapped. We must apply these interchanges consistently to each candidate equation and then see which equation becomes true under normal arithmetic rules. Given Data / Assumptions: We interchange "+" and "/". We interchange the digits 2 and 4 wherever they appear. After the interchange we evaluate the equations with usual precedence. The options to test are: 4/2+3 = 6 2+4/3 = 3 4+2/6 = 1.5 2+4/6 = 8 Concept / Approach: For each option we rewrite the left hand side by swapping all instances of 2 with 4 and all instances of "+" with "/". Then we compute the new left side and compare it with the right side. Only one option will yield a true statement. It is important to apply both the sign swap and the number swap consistently within each expression. Step-by-Step Solution: Option (a): 4/2+3 = 6. Swap 2 and 4 to get 2/4+3, then swap "+" and "/" to get 2+4/3. The new equation is 2 + 4/3 = 6, whose left side is 2 + 1.333... = 3.333..., not 6, so it is false. Option (b): 2+4/3 = 3. Swap 2 and 4 to get 4+2/3, then swap signs to get 4/2+3. The new equation is 4/2 + 3 = 3, whose left side is 2 + 3 = 5, not 3, so it is false. Option (c): 4+2/6 = 1.5. Swap 2 and 4 to get 2+4/6, then swap signs to get 2/4+6. The new equation is 2/4 + 6 = 1.5, whose left side is 0.5 + 6 = 6.5, not 1.5, so it is false. Option (d): 2+4/6 = 8. Swap 2 and 4 to get 4+2/6, then swap signs to get 4/2+6. The new equation is 4/2 + 6 = 8. The left side is 2 + 6 = 8, so this equation is true. Verification / Alternative check: We can double check option (d) directly. After the full interchange, no other option gives an integer on the left side equal to its right side. Only option (d) gives exact equality, so it must be the unique correct choice. Why Other Options Are Wrong: Options (a), (b) and (c) each fail because their transformed left-hands do not match their right-hands. Their left sides evaluate to 3.333..., 5 and 6.5 respectively, while the right sides remain 6, 3 and 1.5. Hence they are clearly incorrect under the specified interchange rules. Common Pitfalls: Many candidates forget to swap both the signs and the numbers, or they swap them only once per equation instead of everywhere they appear. Another source of error is performing the swaps in the wrong order for individual symbols. The safest method is to first swap the digits, then swap the operators, and only after that evaluate the expression. Final Answer: Only the equation that becomes 4/2 + 6 = 8 after the specified interchanges is correct, so the correct choice is 2+4/6 = 8.

Question 16

In a coded arithmetic system, "+" means "minus", "x" means "divided by", "÷" means "plus" and "-" means "multiplied by". Using this code, what is the value of 81 x 9 + 10 – 6 ÷ 5?

Accepted Answer

Introduction / Context: This is another operator coding problem. The four standard arithmetic signs are reassigned to different operations, and the coded expression 81 x 9 + 10 – 6 ÷ 5 must be interpreted accordingly. The final value depends on both correctly decoding the operators and then applying usual precedence rules for multiplication, division, addition and subtraction. Given Data / Assumptions: "+" means "minus". "x" means "divided by". "÷" means "plus". "-" means "multiplied by". The coded expression is 81 x 9 + 10 – 6 ÷ 5. After decoding, we follow standard arithmetic precedence. Concept / Approach: We translate each occurrence of the coded symbols into the correct arithmetic operation. In this case, "x" becomes "/", "+" becomes "-", "–" becomes "*", and "÷" becomes "+". Then we evaluate the resulting expression, carrying out division and multiplication first, followed by addition and subtraction from left to right. Careful handling of negative values is essential. Step-by-Step Solution: Step 1: Decode the operators in 81 x 9 + 10 – 6 ÷ 5. "x" becomes "/", "+" becomes "-", "–" becomes "*", and "÷" becomes "+". Step 2: The decoded expression is 81 / 9 - 10 * 6 + 5. Step 3: Apply precedence. First evaluate 81 / 9 = 9. Step 4: Next compute 10 * 6 = 60. Step 5: Now the expression is 9 - 60 + 5. Step 6: Evaluate from left to right: 9 - 60 = -51. Step 7: Then -51 + 5 = -46. Verification / Alternative check: We can verify by recomputing the intermediate results. The division 81 / 9 gives 9, and the product 10 * 6 gives 60. Substituting these values yields 9 - 60 + 5, which clearly equals -51 + 5 = -46. This matches the option -46 exactly. Why Other Options Are Wrong: 46 and 47 would require the overall sign or one intermediate sign to be reversed. -45 would require the difference between -51 and +5 to be miscalculated as -45 instead of -46. The option 0 would suggest complete cancellation, which does not occur here. Each of these wrong options corresponds to a typical small error in either decoding or arithmetic. Common Pitfalls: The most common errors are mixing up the new meanings of "+" and "-", or performing the addition before the multiplication and division are complete. Another frequent issue is sign mistakes when combining a positive and a negative number. Writing the decoded expression clearly and evaluating stepwise reduces these risks. Final Answer: Hence, after decoding the operators and simplifying, the value of 81 x 9 + 10 – 6 ÷ 5 is -46.

Question 17

In the following equation, two of the arithmetic signs are interchanged so that the equation becomes correct: 6 + 8 ÷ 4 – 4 = 8. Which two signs must be interchanged?

Accepted Answer

Introduction / Context: This question gives an equation that is not true as it stands. You are allowed to interchange exactly two operation signs in order to turn the equation into a correct statement. This type of problem checks your understanding of how each sign affects the value of an expression and how the order of operations influences the final result. Given Data / Assumptions: The original equation is 6 + 8 ÷ 4 – 4 = 8. All numbers remain fixed; only two signs can swap positions. We must apply standard precedence rules during evaluation. The main signs that can be interchanged are "+", "÷" and "–". Concept / Approach: We first evaluate the original left side to see how far it is from 8. Then, for each candidate pair of signs, we swap their positions in the expression and re-evaluate. The correct answer is the pair whose interchange produces a left hand side equal to 8. It is important to respect the precedence of division over addition and subtraction in each case. Step-by-Step Solution: Step 1: Evaluate the original left side: 6 + 8 ÷ 4 – 4. Compute 8 ÷ 4 = 2, so the expression is 6 + 2 – 4 = 8 – 4 = 4, which is not 8. Step 2: Consider interchanging "+" and "–". The expression becomes 6 – 8 ÷ 4 + 4. Step 3: Evaluate 8 ÷ 4 = 2. Step 4: Now the left side is 6 – 2 + 4. Step 5: Perform subtraction and addition from left to right: 6 – 2 = 4, then 4 + 4 = 8. Step 6: Now the left side equals the right side, 8, so this interchange works. Verification / Alternative check: We can briefly test the other possible swaps. Swapping "÷" and "+" or "÷" and "–" yields expressions whose left sides do not evaluate to 8. Swapping "divide and equal to" or introducing multiplication instead of plus also fails to balance the equation. Therefore, only the swap of plus and minus gives a valid equality. Why Other Options Are Wrong: "Divide and equal to" would move the equality sign into the arithmetic part, which does not make sense in standard algebra. "Divide and plus" or "divide and minus" change the structure so that the left side becomes either too large or too small compared with 8. "Multiply and plus" introduces multiplication where it was not intended and again fails to produce 8. Common Pitfalls: A common mistake is to ignore the order of operations and simply compute the expression strictly left to right after the swap. Another is to think that numbers can also be rearranged, which the question does not allow. Only the positions of two operation signs may be interchanged, and all other parts must be left untouched. Final Answer: So, by interchanging the plus and minus signs, the equation becomes correct. The required pair is plus and minus.

Question 18

Consider the following number pattern: 4 x 9 x 3 = 4 and 5 x 3 x 1 = 3. Using the same hidden rule, what is the value of 9 x 9 x 7 = ?

Accepted Answer

Introduction / Context: This question is a classic number puzzle where the usual meaning of the symbol x is hidden behind a special rule. Two sample equations are already solved for you, and you must decode the underlying pattern or operation that converts three given numbers into a single result. Once that rule is understood, you can apply it to a new triple of numbers and determine the missing value from the options. Such questions test a student's ability to detect patterns, experiment with different operations, and reason logically rather than performing standard arithmetic only. Given Data / Assumptions: - The expression 4 x 9 x 3 has been defined to give 4 as the result. - The expression 5 x 3 x 1 has been defined to give 3 as the result. - We assume the hidden rule is the same in both examples and also applies to 9 x 9 x 7. - The usual priority of operations is not relevant here; instead, a special formula is being used. Concept / Approach: The key idea is to search for a consistent relationship that uses all three numbers in each expression. For puzzles like this, we often experiment with combinations of addition, subtraction, and multiplication. After trying simple patterns, we can arrive at a rule that works for both examples: - Multiply the first and second numbers. - Multiply the second and third numbers. - Subtract the second product from the first product. - Subtract twice the first number. - Finally, add the third number. This rule can be written in a compact form as: Result = (first * second) - (second * third) - 2 * first + third Step-by-Step Solution: Step 1: Use the rule for 4 x 9 x 3. First = 4, Second = 9, Third = 3. Compute first * second = 4 * 9 = 36. Compute second * third = 9 * 3 = 27. Subtract these: 36 - 27 = 9. Subtract twice the first number: 9 - 2 * 4 = 9 - 8 = 1. Add the third number: 1 + 3 = 4, which matches the given result. Step 2: Use the same rule for 5 x 3 x 1. First = 5, Second = 3, Third = 1. first * second = 5 * 3 = 15. second * third = 3 * 1 = 3. Difference: 15 - 3 = 12. Subtract twice the first: 12 - 2 * 5 = 12 - 10 = 2. Add the third: 2 + 1 = 3, which matches the second given result. Step 3: Apply the rule to 9 x 9 x 7. First = 9, Second = 9, Third = 7. first * second = 9 * 9 = 81. second * third = 9 * 7 = 63. Difference: 81 - 63 = 18. Subtract twice the first: 18 - 2 * 9 = 18 - 18 = 0. Add the third: 0 + 7 = 7. Verification / Alternative check: After finding a rule that fits both worked examples and produces an integer for the unknown case, we must confirm consistency. The chosen rule produces exactly the known outputs 4 and 3, and also gives a neat, whole number 7 for the new triple. If we tried simpler guesses such as sums, averages, or basic products, they would not match all given results. That supports the validity of this more detailed pattern for the purposes of this puzzle. Why Other Options Are Wrong: Values 5, 6, and 9 do not satisfy the discovered rule. If we plug 9 x 9 x 7 into the formula and force the answer to be 5 or 6 or 9, at least one of the sample equations (4 x 9 x 3 or 5 x 3 x 1) would no longer match its stated result. Hence, those options are inconsistent with the established pattern and must be rejected. Common Pitfalls: Many learners focus only on very simple patterns such as a + b - c, a * b + c, or counting digits. Those attempts quickly fail when tested against both examples. Another common mistake is to try to use standard arithmetic precedence, treating x as usual multiplication and expecting equality in the normal sense. However, in such reasoning questions, x is usually a placeholder for a hidden operation, not ordinary multiplication. Final Answer: Using the consistent hidden rule for all three triples, the value of 9 x 9 x 7 is 7.

Question 19

In a certain code language, "+" represents "×", "−" represents "+", "×" represents "÷" and "÷" represents "−". What is the value of 25 + 10 ÷ 240 × 16 under this coding?

Accepted Answer

Introduction / Context: This problem belongs to the coded mathematical operations category. The usual symbols for addition, subtraction, multiplication, and division are replaced with different meanings. Your task is to carefully decode each symbol, rewrite the expression using actual operations, and then evaluate the resulting arithmetic expression. These questions check attention to detail, correct use of operator precedence, and the ability to handle symbolic substitutions without confusion. Given Data / Assumptions: - In the code language, the symbol "+" means multiplication. - The symbol "−" means addition, although it does not appear in this particular expression. - The symbol "×" means division. - The symbol "÷" means subtraction. - The coded expression to evaluate is: 25 + 10 ÷ 240 × 16. - We assume the usual arithmetic precedence once the expression is decoded: division and multiplication before addition and subtraction, evaluated from left to right at the same level. Concept / Approach: The correct approach is to translate each symbol into its real meaning one by one, while preserving the order of numbers. After this translation, we obtain a standard arithmetic expression that can be evaluated using normal rules. It is crucial not to mix up the original symbols with their coded meanings and not to apply the coded operations directly on the original expression without first rewriting it. Step-by-Step Solution: Step 1: Replace each symbol with its true operation. "+" represents "×", so 25 + 10 becomes 25 × 10. "÷" represents "−", so 10 ÷ 240 becomes 10 − 240. "×" represents "÷", so 240 × 16 becomes 240 ÷ 16. The fully decoded expression is: 25 × 10 − 240 ÷ 16. Step 2: Apply operator precedence. First compute the multiplication: 25 × 10 = 250. Then compute the division: 240 ÷ 16 = 15. Now the expression becomes: 250 − 15. Step 3: Perform the final subtraction. 250 − 15 = 235. Verification / Alternative check: We can quickly verify by re-reading the code mapping. If "+" were misread as addition instead of multiplication, we would obtain a completely different expression and answer. Similarly, if "÷" were misread as division rather than subtraction, the second operation would be incorrect. Confirming that each symbol translation aligns exactly with the problem statement and redoing the arithmetic confirms that 235 is stable as the correct value. Why Other Options Are Wrong: Option 85 would arise if a student divided 240 by 16 first and then used an incorrect combination of operations, perhaps treating "+" as addition and "÷" as division. Option 12 or 20 may appear if the order of operations is ignored and the expression is evaluated strictly left to right or if some of the coded meanings are mixed up. None of these approaches respect both the correct mapping and the standard precedence rules, so they lead to wrong results. Common Pitfalls: Typical mistakes include applying the code backwards (for example, using "+" as addition instead of multiplication), forgetting to respect operator precedence after decoding, or partially decoding the expression and then trying to evaluate it in an inconsistent way. Students should always rewrite the entire expression using normal symbols first and only then perform arithmetic. Final Answer: After decoding the expression and evaluating it correctly, the value of 25 + 10 ÷ 240 × 16 is 235.

Question 20

If "+" means "minus", "×" means "divided by", "÷" means "plus" and "−" means "multiplied by", then what is the value of 38 × 2 − 6 + 19 ÷ 35?

Accepted Answer

Introduction / Context: This question uses a substitution code for arithmetic symbols. Each familiar operator sign is reassigned a new meaning, and you must decode the expression before you can evaluate it. The goal is to check whether the learner can carefully translate and then correctly apply the rules of arithmetic without being misled by the usual appearance of the symbols. Given Data / Assumptions: - "+" now means subtraction (minus). - "×" now means division. - "÷" now means addition. - "−" now means multiplication. - The coded expression is: 38 × 2 − 6 + 19 ÷ 35. - After decoding, we use normal arithmetic precedence: division and multiplication first, followed by addition and subtraction from left to right. Concept / Approach: The method is straightforward. First, translate each coded symbol into its actual arithmetic operation while keeping the order of the numbers unchanged. This converts the coded expression into a standard numerical expression. Then, evaluate step by step, ensuring that multiplication and division are handled before addition and subtraction, just as in standard BODMAS or PEMDAS rules. Step-by-Step Solution: Step 1: Decode each symbol. "×" means "divided by", so 38 × 2 becomes 38 ÷ 2. "−" means "multiplied by", so − 6 becomes × 6. "+" means "minus", so + 19 becomes − 19. "÷" means "plus", so 19 ÷ 35 becomes 19 + 35. Putting this together, the full decoded expression is: 38 ÷ 2 × 6 − 19 + 35. Step 2: Apply division and multiplication from left to right. 38 ÷ 2 = 19. Then 19 × 6 = 114. Now the expression reduces to: 114 − 19 + 35. Step 3: Perform the remaining addition and subtraction from left to right. 114 − 19 = 95. 95 + 35 = 130. Verification / Alternative check: Rechecking the symbol mapping confirms that there is no ambiguity. The converted expression 38 ÷ 2 × 6 − 19 + 35 is standard arithmetic. A quick mental check shows that 38 ÷ 2 is 19, and 19 × 6 is 114, which makes our intermediate steps reliable. Finally, subtracting 19 and adding 35 gives 130, which fits cleanly with the available options and uses all the numbers exactly once, confirming consistency. Why Other Options Are Wrong: An answer like 140 or 135 would typically result from either ignoring operator precedence or misreading one of the coded symbols, for example, treating "+" as addition instead of subtraction or forgetting that "−" indicates multiplication. Option 145 would similarly require incorrect ordering of operations or mixing up addition and subtraction after the initial computations. Since none of these alternatives respect both the substitutions and the precedence rules, they cannot be correct. Common Pitfalls: Students may be tempted to solve the expression as written, with the old meanings of the symbols, which will give a completely different result. Another common error is decoding the operations correctly but then evaluating strictly from left to right without giving priority to division and multiplication. To avoid these mistakes, always rewrite the coded expression fully using real operations first, then apply standard arithmetic rules. Final Answer: The correctly decoded and evaluated value of 38 × 2 − 6 + 19 ÷ 35 is 130.

Mathematical Operations Questions