Problem 7 - Fundamentals of Mathematics

Question

What part of \(1 \frac{1}{4}\) is \(1 \frac{7}{8} ?\)

Step-by-Step Solution

Answer

\(1 \frac{7}{8}\) is \(\frac{3}{2}\) or \(1 \frac{1}{2}\) of \(1 \frac{1}{4}\).

Step 1: Convert Mixed Numbers to Improper Fractions

A mixed number consists of a whole number and a fraction. To make calculations easier, convert the mixed numbers to improper fractions. For \(1 \frac{1}{4}\): Multiply the whole number (1) by the denominator (4) and add the numerator (1). This gives \(1 \times 4 + 1 = 5\). So, \(1 \frac{1}{4}\) converts to \(\frac{5}{4}\).For \(1 \frac{7}{8}\): Multiply the whole number (1) by the denominator (8) and add the numerator (7). This gives \(1 \times 8 + 7 = 15\). So, \(1 \frac{7}{8}\) converts to \(\frac{15}{8}\).

Step 2: Divide the Improper Fractions

To find what part of \(1 \frac{1}{4}\) is \(1 \frac{7}{8}\), divide \(\frac{15}{8}\) by \(\frac{5}{4}\).Division of fractions is equivalent to multiplying the first fraction \(\frac{15}{8}\) by the reciprocal of the second fraction: \(\frac{5}{4}\) becomes \(\frac{4}{5}\).

Step 3: Multiply the Fractions

Multiply \(\frac{15}{8}\) by \(\frac{4}{5}\):\[ \frac{15}{8} \times \frac{4}{5} = \frac{15 \times 4}{8 \times 5} = \frac{60}{40} \]

Step 4: Simplify the Result

The fraction obtained from the multiplication is \(\frac{60}{40}\). Simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 20:\[ \frac{60 \div 20}{40 \div 20} = \frac{3}{2} \]

Step 5: Convert to Mixed Number if Preferred

Though not necessary, if you prefer, convert \(\frac{3}{2}\) to a mixed number. Divide 3 by 2 to get 1 remainder 1, so \(\frac{3}{2}\) as a mixed number is \(1 \frac{1}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are used to represent quantities that include both an integral part and a fractional part. Understanding how to work with mixed numbers is key in many mathematical operations involving fractions.
To convert a mixed number to an improper fraction, you follow a simple method:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to this product.
• Place this sum over the original denominator.

This process helps simplify calculations, such as addition, subtraction, or division of mixed numbers.For example, the mixed number \(1 \frac{1}{4}\) is converted to the improper fraction \(\frac{5}{4}\) by multiplying 1 by 4 and adding 1, resulting in 5. Similarly, \(1 \frac{7}{8}\) becomes \(\frac{15}{8}\) by multiplying 1 by 8 and adding 7.

Fraction Division

Fraction division involves dividing one fraction by another, which can be made simpler by multiplying by the reciprocal.
When dividing fractions, follow these steps:

• Determine the reciprocal of the divisor (flip the numerator and denominator).
• Multiply the dividend by this reciprocal.

The process of using reciprocals makes fraction division straightforward, converting it into a multiplication problem instead. In the example problem, to divide \(\frac{15}{8}\) by \(\frac{5}{4}\), convert \(\frac{5}{4}\) into \(\frac{4}{5}\) (the reciprocal) and multiply: \(\frac{15}{8} \times \frac{4}{5}\).
This multiplication yields \(\frac{60}{40}\), simplifying the division process.

Greatest Common Divisor

The greatest common divisor (GCD) is an important concept in simplifying fractions. It is the largest integer that divides both the numerator and denominator without leaving a remainder.
To simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

In our context, after multiplying fractions in the division problem, we obtained \(\frac{60}{40}\). The GCD of 60 and 40 is 20. Dividing both the numerator and the denominator by 20 simplifies the fraction to \(\frac{3}{2}\). Simplifying fractions makes them easier to understand and compare.