Select the correct combination of mathematical signs to sequentially replace the * signs and balance the given equation. 16 * 3 * 2 * 11 * 9 = 22
× ÷ − +
Finding the Correct Mathematical Signs for Equation Balancing
The problem asks us to find the sequence of mathematical operations ($\times, \div, +, -$) that, when substituted for the asterisks (*) in the equation $16 * 3 * 2 * 11 * 9 = 22$, makes the equation true. We need to test each option provided.
We must follow the standard order of operations (often remembered by acronyms like BODMAS or PEMDAS) when evaluating the expressions: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
Testing Option 1: $\times, \div, +, -$
Let's substitute the signs from Option 1 into the equation:
Equation: $16 \times 3 \div 2 + 11 - 9$
Following the order of operations:
- Perform Multiplication and Division from left to right:
- $16 \times 3 = 48$
- $48 \div 2 = 24$
- The expression becomes: $24 + 11 - 9$
- Perform Addition and Subtraction from left to right:
- $24 + 11 = 35$
- $35 - 9 = 26$
Result: $26$. This does not equal the target value of $22$. So, Option 1 is incorrect.
Testing Option 2: $\times, \div, -, +$
Let's substitute the signs from Option 2 into the equation:
Equation: $16 \times 3 \div 2 - 11 + 9$
Following the order of operations:
- Perform Multiplication and Division from left to right:
- $16 \times 3 = 48$
- $48 \div 2 = 24$
- The expression becomes: $24 - 11 + 9$
- Perform Addition and Subtraction from left to right:
- $24 - 11 = 13$
- $13 + 9 = 22$
Result: $22$. This equals the target value of $22$. So, Option 2 is correct.
Testing Option 3: $\div, \times, +, +$
Let's substitute the signs from Option 3 into the equation:
Equation: $16 \div 3 \times 2 + 11 + 9$
Following the order of operations:
- Perform Multiplication and Division from left to right:
- $16 \div 3 = \frac{16}{3}$
- $\frac{16}{3} \times 2 = \frac{32}{3}$
- The expression becomes: $\frac{32}{3} + 11 + 9$
- Perform Addition from left to right:
- $\frac{32}{3} + 11 = \frac{32}{3} + \frac{33}{3} = \frac{65}{3}$
- $\frac{65}{3} + 9 = \frac{65}{3} + \frac{27}{3} = \frac{92}{3}$
Result: $\frac{92}{3}$. This does not equal the target value of $22$. So, Option 3 is incorrect.
Testing Option 4: $\div, +, \times, \div$
Let's substitute the signs from Option 4 into the equation:
Equation: $16 \div 3 + 2 \times 11 \div 9$
Following the order of operations:
- Perform Multiplication and Division from left to right:
- $16 \div 3 = \frac{16}{3}$
- $2 \times 11 = 22$
- $22 \div 9 = \frac{22}{9}$
- The expression becomes: $\frac{16}{3} + \frac{22}{9}$
- Perform Addition:
- Find a common denominator (which is 9): $\frac{16 \times 3}{3 \times 3} + \frac{22}{9} = \frac{48}{9} + \frac{22}{9}$
- $\frac{48}{9} + \frac{22}{9} = \frac{70}{9}$
Result: $\frac{70}{9}$. This does not equal the target value of $22$. So, Option 4 is incorrect.
Conclusion on Mathematical Signs
Based on the tests, only the sequence of signs $\times, \div, -, +$ from Option 2 balances the given equation $16 * 3 * 2 * 11 * 9 = 22$, resulting in $22 = 22$.
| Option | Signs | Equation | Calculation Steps (BODMAS) | Result | Balances? |
|---|---|---|---|---|---|
| 1 | $\times, \div, +, -$ | $16 \times 3 \div 2 + 11 - 9$ | $48 \div 2 + 11 - 9$ $24 + 11 - 9$ $35 - 9$ $26$ |
$26$ | No |
| 2 | $\times, \div, -, +$ | $16 \times 3 \div 2 - 11 + 9$ | $48 \div 2 - 11 + 9$ $24 - 11 + 9$ $13 + 9$ $22$ |
$22$ | Yes |
| 3 | $\div, \times, +, +$ | $16 \div 3 \times 2 + 11 + 9$ | $\frac{16}{3} \times 2 + 11 + 9$ $\frac{32}{3} + 11 + 9$ $\frac{32}{3} + 20$ $\frac{92}{3}$ |
$\frac{92}{3}$ | No |
| 4 | $\div, +, \times, \div$ | $16 \div 3 + 2 \times 11 \div 9$ | $\frac{16}{3} + 22 \div 9$ $\frac{16}{3} + \frac{22}{9}$ $\frac{48}{9} + \frac{22}{9}$ $\frac{70}{9}$ |
$\frac{70}{9}$ | No |
Revision Table: Equation Balancing Concepts
| Concept | Description | Importance in Equation Balancing |
|---|---|---|
| Mathematical Operators | Symbols like +, -, $\times$, $\div$ representing operations. | Correctly identifying and using these is fundamental. |
| Order of Operations (BODMAS/PEMDAS) | Rules for the sequence in which operations are performed (Brackets, Orders, Division/Multiplication, Addition/Subtraction). | Crucial for evaluating expressions correctly; incorrect order leads to wrong results. |
| Substitution | Replacing placeholders (* in this case) with the given operators. | The first step in testing each option. |
| Evaluation | Calculating the value of the expression after substitution, following the order of operations. | The core process of checking if the equation balances. |
Additional Information on Solving Equation Balancing Problems
Equation balancing problems are common in logical reasoning and quantitative aptitude tests. They assess your understanding of basic arithmetic operations and the order of operations.
- Always write down the equation with the substituted signs clearly.
- Perform calculations step-by-step, strictly following the BODMAS/PEMDAS rule.
- Be careful with division, especially if it results in fractions. Sometimes, fractions will simplify later in the calculation.
- Test each option systematically until you find the one that satisfies the equation.
- Practice with different combinations of numbers and operators to improve speed and accuracy.
Important Questions from Logical Puzzle
Which two numbers should be interchanged to make the given equation correct?
9 + 7 × 5 – 18 ÷ 2 = 3 × 4 – 10 + 45 ÷ 5Select the correct combination of mathematical signs that can sequentially replace the * signs and balance the equation.
60 * 2 * 3 * 6 * 5 * 43
Which of the following interchange of numbers and mathematical signs would make the given equation correct?
30 ÷ 6 × 4 + 15 - 35 = 25
Which two signs need to be interchanged to make the following equation correct?
23 + 84 ÷ 14 × 8 − 3 = 5
Select the correct combination of mathematical signs that can sequentially replace the * signs and make the equation correct.
68 * 138* 23 * 54 * 20
