Problem 55 - Prealgebra

Question

Keri has \(\frac{7}{8}\) gallon of gasoline left. Her mower uses \(\frac{1}{6}\) gallon to cut an average yard. How many average yards can she mow?

Step-by-Step Solution

Answer

Keri can mow 5 full average yards and have enough gas leftover for a quarter of another yard.

Step 1: Understand the problem

Keri needs to know how many full units of \( \frac{1}{6} \) gallon she can fit into \( \frac{7}{8} \) gallon. Thus, we have to divide these two fractions.

Step 2: Set up the division

To find out how many \( \frac{1}{6} \) gallons are in \( \frac{7}{8} \) gallons, we need to perform the division \( \frac{7}{8} \div \frac{1}{6} \). Remember, dividing by a fraction is the same as multiplying by its reciprocal.

Step 3: Multiply by the reciprocal

Find the reciprocal of \( \frac{1}{6} \), which is \( 6 \). Multiply \( \frac{7}{8} \) by \( 6 \) to solve the division: \( \frac{7}{8} \times 6 = \frac{7 \times 6}{8} \).

Step 4: Simplify the fraction

Calculate \( 7 \times 6 = 42 \) so the multiplication gives \( \frac{42}{8} \). Simplify this fraction by finding the greatest common divisor (GCD) of 42 and 8, which is 2. So, \( \frac{42}{8} = \frac{21}{4} \).

Step 5: Convert to a mixed number

Convert \( \frac{21}{4} \) into a mixed number. When you divide 21 by 4, it equals 5 with a remainder of 1. This means \( \frac{21}{4} = 5 \frac{1}{4} \).

Key Concepts

Reciprocal

In the world of fractions, the reciprocal is a crucial concept to grasp. It refers to flipping a fraction upside down. This means the numerator and the denominator swap places. The reciprocal of a fraction like \( \frac{1}{6} \) is straightforwardly \( 6 \). We find the reciprocal by reversing the fraction, where the bottom number (denominator) becomes the top number (numerator), and vice versa.
Why is this useful? When dividing fractions, we actually multiply by the reciprocal. So, \( \frac{7}{8} \div \frac{1}{6} \) turns into \( \frac{7}{8} \times 6 \). This simplifies the operation and makes the math easier to handle.
Remember, finding a reciprocal only works for non-zero numbers, as dividing by zero is undefined in mathematics. Keep practicing this concept by flipping various fractions, and you'll find division involving fractions much easier!

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, which is the largest number that divides both evenly. Divide both parts of the fraction by this GCD.
Take the fraction \( \frac{42}{8} \) from our example. The GCD here is 2. By dividing the numerator (42) and the denominator (8) by 2, we simplify it to \( \frac{21}{4} \). Simplifying not only makes fractions look nicer but also helps in further calculations.
Simplification is essential because it helps reveal the true value of a fraction, making it less cumbersome and more intuitive. Practice it regularly, and turning complex fractions into neat ones will become second nature.

Mixed Numbers

Mixed numbers combine whole numbers and fractions, turning improper fractions easier to read and use. When a fraction's numerator is greater than the denominator, it's called an improper fraction. Converting such fractions into mixed numbers is a handy skill.
In our example, \( \frac{21}{4} \) is an improper fraction. To turn it into a mixed number, divide 21 by 4. The result is 5 with a remainder of 1. So \( \frac{21}{4} = 5 \frac{1}{4} \).
Each mixed number reflects a more intuitive representation of whole values plus a fraction. This format is particularly helpful in real-life situations, such as measuring and dividing items, because it shifts abstract math into manageable pieces. Understanding and converting between improper fractions and mixed numbers is crucial for math fluency.