Problem 43 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\frac{51}{17}+7-2 \cdot 5 \cdot\left(\frac{12}{3}\right)\)

Step-by-Step Solution

Answer

The result is -30.

Step 1: Simplify the Fraction

Start by simplifying the fraction \(\frac{51}{17}\). Divide 51 by 17 to get 3.

Step 2: Calculate Inside the Parentheses

Next, calculate \(\frac{12}{3}\), which simplifies to 4.

Step 3: Multiply Values

Now calculate the multiplication part: \(2 \cdot 5 \cdot 4\). First, multiply 2 and 5 to get 10. Then, multiply 10 and 4 to get 40.

Step 4: Substitute Values Back into the Expression

Substitute the simplified values back into the expression: \(3 + 7 - 40\).

Step 5: Final Calculation

Finally, calculate the expression: \(3 + 7 = 10\) and then \(10 - 40 = -30\).

Step 6: Verify with Calculator

Check the calculation using a calculator to confirm that the steps were performed correctly and the final result is indeed \(-30\).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them first can make the rest of the problem easier to handle. Simplifying a fraction means reducing it to its simplest form. This involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common factor (GCF).

• For example, in the fraction \( \frac{51}{17} \), both 51 and 17 are divisible by 17. Performing the division \(51 \div 17\) gives us 3.
• Similarly, if you have \( \frac{12}{3} \), you divide 12 by 3, resulting in 4.

In problems where fractions appear, always check if you can simplify them first before proceeding with other operations.

Parentheses in Math

Parentheses play a crucial role in mathematical expressions. They indicate which operations should be performed first. The order of operations dictates that you perform any calculations inside parentheses before moving on to multiplication, division, addition, or subtraction. This rule is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• In our example, \( \left( \frac{12}{3} \right) \) is calculated first, yielding 4.
• Ensure you always resolve expressions within the parentheses to follow this reliable order of operations.

Neglecting this step can lead to incorrect results, as the operations outside the parentheses depend on the values derived from it.

Multiplication of Numbers

Once you have simplified fractions and resolved any expressions within parentheses, you can move on to multiplication. Multiplication in an expression without parentheses should follow after solving any in-bracket operations. This is part of the order of operations, ensuring calculations proceed logically and accurately.

• In the provided exercise, you have \( 2 \cdot 5 \cdot 4 \) after simplifying \( \frac{12}{3} \) to 4.
• First, calculate \( 2 \cdot 5 \) which equals 10.
• Then multiply 10 by 4, resulting in 40.

By handling multiplication correctly, subsequent addition and subtraction steps become straightforward, leading to accurate solutions.