Balancing Mathematical Equations with Signs

This problem asks us to find the correct sequence of mathematical signs that, when placed in the given equation, makes both sides equal. We are given an equation with placeholder symbols (*) and several options containing sequences of standard mathematical operations and an equals sign.

Understanding the Problem

The equation is 15 * 5 * 16 * 15 * 2 * 61. There are five '*' symbols. The options provide five symbols: the first four are mathematical operations ($\div$, $\times$, $+$, $-$), and the fifth symbol is $=$. This means we need to replace the first four '*' signs with the operations from an option and check if the calculation on the left side equals the number on the right side (61 in this case).

Step-by-Step Solution: Testing Options

To solve this, we will test each option provided. For each option, we will replace the '*' symbols with the given signs in order and perform the calculation on the left side of the equation. We must follow the standard order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Testing Option 1: $\div$, $\times$, +, $\div$, =

Let's substitute the signs into the equation:

$15 \div 5 \times 16 + 15 \div 2 = 61$

Now, let's perform the calculation following the order of operations:

• First, Division: $15 \div 5 = 3$ and $15 \div 2 = 7.5$. The equation becomes:
• $3 \times 16 + 7.5 = 61$
• Next, Multiplication: $3 \times 16 = 48$. The equation becomes:
• $48 + 7.5 = 61$
• Finally, Addition: $48 + 7.5 = 55.5$. The equation becomes:
• $55.5 = 61$

The left side ($55.5$) is not equal to the right side ($61$). So, Option 1 is incorrect.

Testing Option 2: $\div$, $\times$, +, -, =

Let's substitute the signs into the equation:

$15 \div 5 \times 16 + 15 - 2 = 61$

Now, let's perform the calculation following the order of operations:

• First, Division: $15 \div 5 = 3$. The equation becomes:
• $3 \times 16 + 15 - 2 = 61$
• Next, Multiplication: $3 \times 16 = 48$. The equation becomes:
• $48 + 15 - 2 = 61$
• Finally, Addition and Subtraction (from left to right): $48 + 15 = 63$, then $63 - 2 = 61$. The equation becomes:
• $61 = 61$

The left side ($61$) is equal to the right side ($61$). So, Option 2 is correct.

Testing Option 3: -, $\times$, +, $\div$, =

Let's substitute the signs into the equation:

$15 - 5 \times 16 + 15 \div 2 = 61$

Now, let's perform the calculation following the order of operations:

• First, Multiplication and Division (from left to right): $5 \times 16 = 80$ and $15 \div 2 = 7.5$. The equation becomes:
• $15 - 80 + 7.5 = 61$
• Finally, Subtraction and Addition (from left to right): $15 - 80 = -65$, then $-65 + 7.5 = -57.5$. The equation becomes:
• $-57.5 = 61$

The left side ($-57.5$) is not equal to the right side ($61$). So, Option 3 is incorrect.

Testing Option 4: $\div$, $\times$, -, $\div$, =

Let's substitute the signs into the equation:

$15 \div 5 \times 16 - 15 \div 2 = 61$

Now, let's perform the calculation following the order of operations:

• First, Division: $15 \div 5 = 3$ and $15 \div 2 = 7.5$. The equation becomes:
• $3 \times 16 - 7.5 = 61$
• Next, Multiplication: $3 \times 16 = 48$. The equation becomes:
• $48 - 7.5 = 61$
• Finally, Subtraction: $48 - 7.5 = 40.5$. The equation becomes:
• $40.5 = 61$

The left side ($40.5$) is not equal to the right side ($61$). So, Option 4 is incorrect.

Conclusion on Correct Option

Based on our testing, only Option 2, with the signs $\div$, $\times$, +, -, =, correctly balances the given equation.

Summary Table of Options Tested

Revision Table: Key Concepts

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. This is crucial for consistently solving equations.

A common acronym for remembering the order is BODMAS or PEMDAS:

• Brackets (or Parentheses)
• Orders (Powers and Square Roots, or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

When you have division and multiplication together, or addition and subtraction together, you perform them from left to right as they appear in the expression.