Problem 40 - Prealgebra

Question

Write a real-world problem that you could solve by subtracting \(2 \frac{1}{8}\) from \(15 \frac{3}{4}\)

Step-by-Step Solution

Answer

The remaining amount of sugar is \(13 \frac{5}{8}\) cups.

Step 1: Understand the Problem

Imagine you are baking a cake and need a specific amount of sugar. You started with a stock of \(15 \frac{3}{4}\) cups of sugar, and after baking, you have used up \(2 \frac{1}{8}\) cups. The problem is to find out how much sugar is left.

Step 2: Convert Mixed Numbers to Improper Fractions

To simplify subtraction, convert the mixed numbers to improper fractions. First, convert \(15 \frac{3}{4}\):\[15 \frac{3}{4} = \frac{(15\times4) + 3}{4} = \frac{60+3}{4} = \frac{63}{4}\].Now convert \(2 \frac{1}{8}\):\[2 \frac{1}{8} = \frac{(2\times8) + 1}{8} = \frac{16+1}{8} = \frac{17}{8}\].

Step 3: Find a Common Denominator

The fractions have different denominators (4 and 8). To subtract, we need a common denominator. The least common denominator of 4 and 8 is 8. Convert \(\frac{63}{4}\) to a fraction with a denominator of 8: \[\frac{63}{4} = \frac{63 \times 2}{4 \times 2} = \frac{126}{8}\].

Step 4: Perform the Subtraction

Subtract the improper fractions: \[\frac{126}{8} - \frac{17}{8} = \frac{126 - 17}{8} = \frac{109}{8}\].

Step 5: Convert the Improper Fraction to a Mixed Number

Divide 109 by 8 to convert the improper fraction back to a mixed number. 109 divided by 8 gives 13 with a remainder of 5. Thus, \(\frac{109}{8}\) is \(13 \frac{5}{8}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are commonly used to represent quantities that are more than a whole but not quite a full additional whole number. For example, in the problem of sugar in the pantry, we start with a mixed number of sugar: \(15 \frac{3}{4}\) cups. This means you have 15 whole cups, plus an additional three-fourths of a cup. Mixed numbers make it easier to visualize and work with larger quantities in everyday life. They are especially useful in cooking and baking, where precise measurements are essential.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is equal to or larger than the denominator (the bottom number). This can be useful for calculations such as subtraction, because it keeps everything in terms of fractions rather than having to work with a whole number and a part of a fraction. In our example, we convert the mixed numbers \(15 \frac{3}{4}\) and \(2 \frac{1}{8}\) into improper fractions to make subtraction easier. This results in \(\frac{63}{4}\) and \(\frac{17}{8}\) respectively. Once in this form, they can be easily compared and subtracted, allowing us to find the remaining sugar.

Common Denominator

A common denominator is crucial when subtracting fractions. It simplifies the process because the denominators must match to perform the subtraction directly. Think of the denominator as dividing a whole into equal parts, so both fractions must have the same sized parts to work together. In this exercise, the denominators are 4 and 8. We find the least common multiple of these, which is 8, to be our common denominator. We then convert \(\frac{63}{4}\) to \(\frac{126}{8}\), making it easy to subtract \(\frac{17}{8}\). This important step ensures a smooth and accurate subtraction.

Real-World Problem

Real-world problems like the one in our exercise help to see the application of mathematical concepts in daily life. Imagine needing to determine how much sugar you have left after some was used for baking. This is a practical situation where subtraction with mixed numbers and improper fractions comes into play. By using these mathematical concepts, you can mathematically determine that from an initial stock of \(15 \frac{3}{4}\) cups of sugar, and after using \(2 \frac{1}{8}\) cups, you would have \(13 \frac{5}{8}\) cups remaining. Understanding such problems and their solutions helps reinforce why these mathematical techniques are valuable.