The correct answer is B. The expression 4 × (125 + 67) means we multiply (125 + 67) by 4, making it 4 times as large as 125 + 67. Choice A would be true if the first expression was (125 + 67) + 4. Choice C would be true if the first expression was (125 + 67) - 4. Choice D reverses the relationship.

The correct answer is C. "The difference of 23 and 15" means (23 - 15), and we multiply 7 by this entire difference: 7 × (23 - 15). Choice A multiplies 7 by 23 first, then subtracts 15. Choice B is the same as A with explicit grouping. Choice D creates the wrong difference and multiplies by the wrong number.

The correct answer is B. Student B correctly shows that 20 - 5 must be calculated first (using parentheses), then the result is multiplied by 4: (20 - 5) × 4. Student A's expression would multiply 5 × 4 first due to order of operations. Student C multiplies 5 × 4 first, then subtracts from 20. Student D multiplies 4 × 20 first, then subtracts 5.

When you see a word problem asking you to translate a process into a mathematical expression, you need to follow the order of operations exactly as described and use parentheses to show which steps happen together.

Let's trace through the baker's process step by step. First, "double the number of chocolate chips" means multiply $$c$$ by 2, giving us $$2c$$. Next, "add 8 vanilla chips" means we add 8 to get $$2c + 8$$. Finally, "divide the total by 3" means we take everything we've calculated so far and divide it by 3. Since we need to divide the entire sum $$2c + 8$$ by 3, we need parentheses: $$(2c + 8) \div 3$$. This makes choice A correct.

Choice B, $$2c + 8 \div 3$$, follows the wrong order of operations. Without parentheses, division happens before addition, so this would mean "double the chocolate chips, then add the result of 8 divided by 3," which isn't what the problem describes. Choice C, $$2(c + 8) \div 3$$, adds 8 to the chocolate chips first, then doubles everything—this reverses the first two steps. Choice D, $$2c + (8 \div 3)$$, divides only the vanilla chips by 3, not the total amount.

The key strategy here is to translate word problems one step at a time in the exact order given, then use parentheses to group operations that should happen together before the next step. Always double-check that your expression matches the sequence described in the problem.

The correct answer is A. The calculation requires first finding the sum of 15 and 9, which needs parentheses: (15 + 9), then subtracting 6 from that result: (15 + 9) - 6. Choice B incorrectly groups 9 - 6 first. Choice C lacks parentheses, so it would be calculated left to right but doesn't emphasize the required grouping. Choice D incorrectly groups 15 - 6 first.

The correct answer is C. Addition is commutative, so 6 + (4 × 9) equals (4 × 9) + 6. Choice A uses the distributive property incorrectly. Choice B changes the grouping so that 6 × 4 is calculated first. Choice D adds 6 to 9 before multiplying by 4, changing the calculation order.

The correct answer is C. To make an expression 3 times larger, we multiply the entire expression by 3: 3 × (45 + 28). Choice A adds 3 to the original expression, making it 3 more, not 3 times larger. Choice B is the same as A due to commutative property. Choice D makes the expression 3 times smaller by dividing by 3.

When comparing expressions without calculating exact values, look for patterns and relationships between the operations. This helps you understand how numbers relate to each other mathematically.

Both expressions start with the same quantity: $$(50 + 25)$$. The key difference is what happens to this sum. In the first expression, $$2 \times(50 + 25)$$, you're multiplying the sum by 2, which doubles it. In the second expression, $$(50 + 25) \div 2$$, you're dividing the sum by 2, which cuts it in half.

Think about it this way: if the sum $$(50 + 25)$$ represents some value, then doubling it gives you twice that value, while halving it gives you half that value. The relationship between "twice something" and "half of something" is that the first is 4 times larger than the second. This is because $$2 \times \text{(original)} = 4 \times \frac{1}{2} \times \text{(original)}$$.

Choice A is wrong because the difference isn't a simple addition of 2. Choice B incorrectly reverses the relationship – the first expression is larger, not smaller. Choice C misses that we're comparing a doubled amount to a halved amount, not just comparing the original to a doubled amount.

When comparing expressions with the same base value but different operations, focus on how those operations relate to each other. Multiplication and division by the same number create predictable ratios – multiplying by 2 versus dividing by 2 always creates a 4-to-1 relationship.

When you see an expression with parentheses and multiple operations, you need to carefully match it to the order of operations described in the word problem. The expression $$5 \times(12 + 8) \div 2$$ tells you exactly what to do step by step.

Following the order of operations (PEMDAS), you first handle what's in parentheses: $$(12 + 8)$$. This gives you the sum of 12 and 8, which equals 20. Next, you multiply that result by 5: $$5 \times 20 = 100$$. Finally, you divide by 2: $$100 \div 2 = 50$$.

Choice D matches this perfectly: "Find the sum of 12 and 8, multiply by 5, then divide by 2." This follows the exact same sequence as the mathematical expression.

Choice A is wrong because it says "Find 5 times 12, add 8" - but the parentheses in the expression clearly show that 12 and 8 must be added first, not that 5 and 12 are multiplied first.

Choice B reverses the order by dividing before multiplying: "divide by 2, then multiply by 5." This would give you $$5 \times(12 + 8 \div 2)$$, which is a different expression entirely.

Choice C breaks up the addition in parentheses, suggesting you multiply 5 by 12 separately, then add "8 divided by 2." This ignores the parentheses completely and changes the meaning.

Remember: parentheses are your roadmap! Whatever is inside parentheses must be calculated first, and the operations outside follow the normal order. Always match the word problem to this mathematical sequence.

Numerical expressions describe calculations using numbers and operations without computing the final answer. When reading expressions like 6 × 12 - 15, it's important to carefully note the operations and any groupings to understand the sequence of steps. Matching words to operations means identifying that '6 packs with 12 each' corresponds to multiplication, and 'gives away 15' to subtraction. Grouping symbols connect directly to the meaning by indicating what to calculate first, such as ensuring multiplication happens before subtraction in this case. A common misconception is thinking that expressions must include parentheses for all operations, but here the order of operations handles multiplication before subtraction without them. Numerical expressions are useful because they provide a clear way to represent real-world situations mathematically. They help us communicate and verify calculations accurately in problems like tracking inventory or resources.