Problem 80 - Algebra 1

Question

Evaluate the expression. $$2-8 \div \frac{-2}{3}$$

Step-by-Step Solution

Answer

The value of the expression \(2 - 8 \div \frac{-2}{3}\) is 14.

Step1: Handle Division first

According to BIDMAS/BODMAS, division should be handled before subtraction. The division given in the equation is \(8 \div \frac{-2}{3}\). To perform this operation, multiply 8 by the reciprocal of \(\frac{-2}{3}\), that is \(\frac{-3}{2}\).

Step 2: Perform the multiplication

Now perform the multiplication, \(8 * \frac{-3}{2}\), which equals to -12.

Step 3: Perform the subtraction

Now, substitute -12 into the original expression in place of \(8 \div \frac{-2}{3}\), we will get \(2 - -12\). Two negatives make a positive, so this becomes \(2 + 12\).

Step 4: Final calculation

Evaluate the final expression \(2 + 12\), which equals 14.

Key Concepts

Division

In mathematical expressions, division is a process of determining how many times one number is contained within another. It is also the inverse operation of multiplication. When approaching division, especially within more complex arithmetic expressions, it’s vital to follow certain rules to ensure accurate results.

• Always look for division symbols or terms that need dividing in your expression.
• Replace the division with a multiplication by the reciprocal of the divisor. For instance, the division \( 8 \div \frac{-2}{3} \) can be transformed into a multiplication \( 8 * \frac{-3}{2} \). Here, \( \frac{-3}{2} \) is the reciprocal of \( \frac{-2}{3} \).
• Carry out the multiplication as per the standard multiplication rules, taking care with negative signs to ensure the result has the right sign (i.e., a negative divided by a positive yields a negative result).

Using these simple steps helps to manage order of operations effectively, and avoids common pitfalls that can arise in division calculations.

Multiplication

Multiplication is a mathematical operation used to scale one number by another. Within the context of an expression, multiplication often follows directly after a division when simplifying. After converting division by a fraction into multiplication, the process becomes more straightforward.

• In the expression, once the division \( 8 \div \frac{-2}{3} \) has been turned into \( 8 * \frac{-3}{2} \), calculate the product.
• To multiply, calculate \( 8 \times \frac{-3}{2} \) by carrying out the operation \( 8 \times -3 \), which results in \(-24\).
• Then divide \(-24\) by 2, resulting in \(-12\). This multiplication and subsequent division mimic the original division process.

Through proper execution of multiplication steps, especially when involving negatives or fractions, the integration of these skills allows for clear and correct final expressions.

BIDMAS/BODMAS

BIDMAS/BODMAS is a crucial guiding acronym for handling the order of operations within an expression.

- **B** stands for Brackets, ensuring anything in parentheses is calculated first.- **I/D** for Indices (or "Orders"), which means exponents or powers.- **D/M** represents Division and Multiplication; these operations are performed from left to right, whichever comes first.- **A/S** for Addition and Subtraction, solved last, also from left to right.

Understanding BIDMAS/BODMAS helps to prevent mistakes when performing calculations. For instance, in the expression \( 2 - 8 \div \frac{-2}{3} \), division is executed before subtraction due to this order. This rule ensures all calculations flow in a logical and predictable sequence. By properly applying BIDMAS/BODMAS, even complex expressions become manageable and straightforward to solve.