Problem 75 - Algebra 1

Question

Evaluate the expression. $$-3 \cdot\left[\left(2 \frac{9}{14}-3 \frac{3}{7}\right) \cdot \frac{28}{11}\right]+5\left(-9 \frac{1}{5}-9\right)$$

Step-by-Step Solution

Answer

The solution to the expression is -106

Step 1: Simplify Inside Parentheses

The first priority is to simplify the operations inside the parentheses. Here, we have two fractions within the parentheses: \(2 \frac{9}{14}\) and \(3 \frac{3}{7}\). Simplify these by converting the mixed numbers to improper fractions. \(2 \frac{9}{14}\) becomes \(\frac{37}{14}\) and \(3 \frac{3}{7}\) becomes \(\frac{24}{7}\). Now subtract the two fractions: \(\frac{37}{14} - \frac{24}{7} = \frac{37}{14} - \frac{48}{14} = -\frac{11}{14}\). Now, simplify the other bracket, turning \(-9 \frac{1}{5}\) to \(\frac{-46}{5}\). So, the expression becomes: \[-3 \cdot \left[ -\frac{11}{14} \cdot \frac{28}{11} \right] +5 \left(-\frac{46}{5} - 9 \right)\]

Step 2: Perform the Multiplication and Division

Next, carry out the multiplication and division in each of the square brackets. The first square bracket becomes \( -3 \cdot 2 = -6 \) and the second square bracket simplifies to \( 5\cdot (-20) = -100 \). The expression now becomes: \[-6 - 100\]

Step 3: Complete the Addition/Subtraction

Complete the subtraction in the expression: \[-6 - 100 = -106\]

Key Concepts

Mixed Numbers to Improper Fractions

When dealing with algebraic expressions that involve mixed numbers, it is essential first to convert these mixed numbers into improper fractions. A mixed number is a combination of a whole number and a proper fraction. To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator of the fraction to this product. The result becomes the numerator of the improper fraction, while the denominator remains the same. For instance, to convert the mixed number \( 2 \frac{9}{14} \) to an improper fraction, multiply \(2\) (the whole part) by \(14\) (the denominator) to get \(28\), and add \(9\) (the numerator) resulting in \(37\). Therefore, the improper fraction is \( \frac{37}{14} \). This conversion simplifies the process of performing operations such as addition, subtraction, multiplication, and division which we often encounter in algebra.

Operations Inside Parentheses

Handling operations inside parentheses is a fundamental skill in algebra. According to the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), calculations within parentheses should be completed first before moving on to exponents, and then multiplication and division, which are treated with equal importance and performed from left to right, followed lastly by addition and subtraction.

When working with expressions inside parentheses, first simplify any operations including fractions or mixed numbers as mentioned before. After simplification, if there are multiple terms within the parentheses, you must then process these terms according to the order of operations. The importance of carefully executing operations inside parentheses cannot be overstated, as failing to do so will likely result in incorrect solutions to otherwise straightforward problems.

Multiplication and Division of Fractions

Multiplication and division of fractions are more straightforward operations than one might expect. To multiply fractions, simply multiply the numerators to find the new numerator and the denominators to find the new denominator. There is no need to find a common denominator when multiplying fractions; for example, \( \frac{3}{4} \times \frac{2}{5} \) becomes \( \frac{3 \times 2}{4 \times 5} = \frac{6}{20} \), which simplifies to \( \frac{3}{10} \).

For division, you 'multiply by the reciprocal'. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Thus, when dividing one fraction by another, like \( \frac{5}{6} \div \frac{1}{2} \), you will multiply \( \frac{5}{6} \) by the reciprocal of \( \frac{1}{2} \) which is \( \frac{2}{1} \) to get \( \frac{5 \times 2}{6 \times 1} = \frac{10}{6} \) which simplifies to \( \frac{5}{3} \). Remember, simplification of fractions is also crucial after multiplication or division to express the result in its simplest form.