To place parentheses to make equation true, you must use grouping symbols to change the order in which operations are performed. Parentheses tell us to calculate inside them first, and this can completely change the value of an expression. This skill is important because many math problems look similar at first, but the placement of parentheses can turn an incorrect equation into a correct one Worth knowing..
Introduction: Why Parentheses Matter
In mathematics, the order of operations matters. Without parentheses, we usually follow the standard order:
- Parentheses or grouping symbols
- Exponents
- Multiplication and division, from left to right
- Addition and subtraction, from left to right
This is often remembered using the phrase PEMDAS or BODMAS.
Parentheses are powerful because they override the normal order. For example:
8 + 2 × 5 = 18
This equation is true if multiplication happens first:
8 + 10 = 18
But if we place parentheses like this:
(8 + 2) × 5 = 50
Now the addition happens first, so the result changes.
That is the basic idea behind learning to place parentheses to make equation true: you adjust the grouping so both sides of the equation have the same value.
What Are Parentheses?
Parentheses are the symbols written like this:
( )
They group numbers or expressions together. Anything inside parentheses is calculated first.
For example:
3 + (4 × 2) = 11
Inside the parentheses:
4 × 2 = 8
Then:
3 + 8 = 11
Without parentheses:
3 + 4 × 2 = 11
In this case, the answer is still 11 because multiplication already comes before addition. But sometimes parentheses are necessary to change the result And it works..
Example:
3 × 4 + 2 = 14
But:
(3 × 4) + 2 = 14
And:
3 × (4 + 2) = 18
The parentheses changed the meaning of the expression.
What Does “Make Equation True” Mean?
An equation has an equals sign. It says that the expression on the left side has the same value as the expression on the right side.
For example:
5 + 3 = 8
This is true because both sides equal 8.
But:
5 + 3 = 9
This is false because 8 is not equal to 9.
When a problem asks you to place parentheses to make equation true, it usually gives you numbers, operations, and an equals sign. Your job is to insert parentheses in the correct places so the equation becomes correct.
Example:
6 + 4 × 5 = 50
At its core, false if we follow the normal order of operations:
6 + 20 = 26
But if we place parentheses around 6 + 4, we get:
(6 + 4) × 5 = 50
Now it is true:
10 × 5 = 50
Step-by-Step Method for Placing Parentheses
Step 1: Read the Whole Equation
Do not rush. Look at the numbers, operations, and the target value on the right side.
For example:
9 - 4 × 2 = 10
The right side is 10. Your goal is to place parentheses so the left side equals 10 Still holds up..
Step 2: Calculate Without Parentheses
First, check what the expression equals using the normal order of operations.
9 - 4 × 2
Multiplication comes first:
4 × 2 = 8
Then:
9 - 8 = 1
So without parentheses, the left side is 1, but the right side is 10.
Step 3: Look for Useful Groupings
Now think about what grouping could change the result.
Try:
(9 - 4) × 2
Inside the parentheses:
9 - 4 = 5
Then:
5 × 2 = 10
So the true equation is:
(9 - 4) × 2 = 10
Step 4: Check Your Answer
Always substitute your parentheses back into the equation and calculate again.
(9 - 4) × 2 = 10
5 × 2 = 10
10 = 10
The equation is true Simple, but easy to overlook..
Common Types of Parentheses Problems
1. Addition and Multiplication Problems
These are common because multiplication normally happens before addition.
Example:
7 + 3 × 5 = 50
Without parentheses:
7 + 15 = 22
Try grouping the addition:
(7 + 3) × 5
10 × 5 = 50
So:
(7 + 3) × 5 = 50
2. Subtraction and Division Problems
Division often comes before subtraction, so parentheses can change the result.
Example:
20 - 12 ÷ 3 = 8
Without parentheses:
20 - 4 = 16
Try grouping the subtraction:
(20 - 12) ÷ 3
8 ÷ 3
That
