Questions: Evaluating Expressions with Grouping Symbols

Question 1

What is the value of (2 + 3) × 4?

• A. 20
• B. 14
• C. 24
• D. 10

Answer: (A)

The parentheses force addition to happen before multiplication. (2 + 3) = 5, then 5 × 4 = 20. Option B (14) is what you get without the parentheses, following default order of operations: 2 + (3 × 4) = 2 + 12 = 14. This is precisely why parentheses exist — to override the default order when you need a different result.

Question 2

What is the value of {3 × [2 + (8 − 6)]}?

• A. 12
• B. 24
• C. 18
• D. 6

Answer: (A)

Work from the innermost group outward. Step 1: (8 − 6) = 2. Rewrite: {3 × [2 + 2]}. Step 2: [2 + 2] = 4. Rewrite: {3 × 4} = 12. A common mistake is to evaluate left to right — 3 × 2 = 6, then 6 + 8 = 14, then 14 − 6 = 8 — which ignores the grouping hierarchy entirely.

Question 3

Parentheses, brackets, and braces all carry the same mathematical instruction: evaluate the contents first, before applying outer operations.

• T. True
• F. False

Answer: True

All three types of grouping symbols mean 'evaluate me first.' The three different shapes exist purely for visual clarity when symbols are nested — using distinct shapes makes it easier to match each opening symbol with its correct closing symbol in complex expressions like {3 × [2 + (8 − 6)]}.

Question 4

When evaluating a nested expression like {2 × [5 + (3 − 1)]}, you should evaluate the outermost group first and work your way inward.

• T. True
• F. False

Answer: False

The rule is the opposite: evaluate the innermost group first and work outward. The innermost group is the one with no further grouping inside it. In {2 × [5 + (3 − 1)]}, start with (3 − 1) = 2, then [5 + 2] = 7, then {2 × 7} = 14. Starting from the outside would leave you with an unresolved inner expression.

Question 5

Why are three different types of grouping symbols — parentheses ( ), brackets [ ], and braces { } — used in mathematics if they all mean the same thing?

Model answer: Different shapes help you match each opening symbol with its correct closing symbol when grouping symbols are nested inside each other. Using three visually distinct shapes makes it easier to read complex nested expressions and verify that every opener is paired with the right closer.

In an expression like {2 × [3 + (4 − 1)]}, the three shapes let your eye immediately find that the innermost ( ) pair encloses 4 − 1, the [ ] pair encloses the addition, and the { } pair encloses the whole multiplication. If all three levels used the same shape, nested expressions would be much harder to parse correctly.