Problem 30 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ 54 \div\left(-2 \frac{4}{7}\right) $$

Step-by-Step Solution

Answer

The quotient is -21.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(-2 \frac{4}{7}\) to an improper fraction. This can be done by multiplying the denominator with the whole number and adding it to the numerator to form the new numerator. Here, \(-2 \frac{4}{7}\) converts to \(-\frac{18}{7}\).

Step 2: Convert Division to Multiplication

We now convert the operation of division to the operation of multiplication by using the concept that 'division by a number is the same as multiplication by its reciprocal'. Reciprocal of \(-\frac{18}{7}\) is \(-\frac{7}{18}\). So, the expression converts to \(54 \times -\frac{7}{18}\).

Step 3: Perform Multiplication

Multiplying the integers, we have \(54 \times -\frac{7}{18} = -21\).

Key Concepts

Understanding Improper Fractions

When dealing with improper fractions, we focus on fractions where the numerator is greater than or equal to the denominator. Essentially, this means the fraction represents a number equal to or greater than 1. For example, if you consider a fraction such as \( \frac{18}{7} \), the numerator 18 is greater than the denominator 7. If you visualize this, it means you have more parts than what one whole can provide, which is why it's "improper."Improper fractions can often be converted to mixed numbers, which typically involves dividing the numerator by the denominator. The quotient becomes the whole number and the remainder the new numerator. However, in operations like fraction division, it's often more helpful to convert mixed numbers into improper fractions.

The Role of Reciprocals

A reciprocal is essentially a flipped fraction. If you have a fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). Reciprocals play a critical role when you're dividing fractions or mixed numbers. The essence here is that dividing by a fraction is the same as multiplying by its reciprocal. For instance, if you are dividing by \(-\frac{18}{7}\), this is the same as multiplying by \(-\frac{7}{18}\). Keep in mind that this flipping works similarly for both positive and negative numbers.

Mixed Numbers Made Simple

Mixed numbers are combinations of whole numbers and proper fractions. For instance, in \(-2 \frac{4}{7}\), "\(-2\)" is the whole number and "\(\frac{4}{7}\)" is the fractional part. They often present a more natural way to express quantities that aren’t whole and are useful in everyday calculations. Converting mixed numbers to improper fractions requires multiplying the whole number by the denominator and adding the numerator. This totals a single larger fractional part that incorporates the whole number into the overall fraction. This step is essential when dividing or multiplying, as it eases the calculation process by having a consistent form.

Simplifying with Integer Multiplication

Multiplying integers involves straightforward calculations but can be tricky when negative numbers are included. In the operation \(54 \times -\frac{7}{18}\), start by multiplying the integer 54 by 7, which gives 378. Then, divide this result by 18. Since you are multiplying by \(-\frac{7}{18}\), the negative sign means the quotient will be negative. When you divide, think of simplifying beforehand if possible to lessen the computation. Importantly, while dividing doesn’t change the sign, it influences the result’s direction on the number line, leading you to \(-21\). Understanding these steps ensures you handle similar problems correctly.