Problem 53 - Fundamentals of Mathematics

Question

Arrange the numbers in order from smallest to largest: \(\frac{11}{32}, \frac{15}{48},\) and \(\frac{7}{16}\).

Step-by-Step Solution

Answer

The order from smallest to largest is \(\frac{15}{48}, \frac{11}{32}, \frac{7}{16}\).

Step 1: Convert to a Common Denominator

To compare the fractions \(\frac{11}{32}, \frac{15}{48}, \frac{7}{16}\), we need to convert them to a common denominator. The least common multiple of 32, 48, and 16 is 96.

Step 2: Convert Each Fraction

Convert each fraction to have a denominator of 96:- For \(\frac{11}{32}\), multiply the numerator and denominator by 3: \(\frac{11 \times 3}{32 \times 3} = \frac{33}{96}\).- For \(\frac{15}{48}\), multiply the numerator and denominator by 2: \(\frac{15 \times 2}{48 \times 2} = \frac{30}{96}\).- For \(\frac{7}{16}\), multiply the numerator and denominator by 6: \(\frac{7 \times 6}{16 \times 6} = \frac{42}{96}\).

Step 3: Compare the Fractions

Now we compare the converted fractions:- \(\frac{30}{96}\)- \(\frac{33}{96}\)- \(\frac{42}{96}\)From smallest to largest: \(\frac{30}{96} < \frac{33}{96} < \frac{42}{96}\).

Step 4: Write the Original Fractions in Order

Now revert to the original fractions according to their order:- The smallest is \(\frac{15}{48}\)- The middle is \(\frac{11}{32}\)- The largest is \(\frac{7}{16}\)Ordered from smallest to largest: \(\frac{15}{48}, \frac{11}{32}, \frac{7}{16}\).

Key Concepts

Common Denominator

To effectively compare fractions, it's necessary to work with a common denominator. Think of a common denominator as a shared foundation that allows us to equitably evaluate diverse fractions. When fractions have the same denominator, you can easily compare them by just looking at their numerators. This process involves adapting each fraction to a shared baseline, facilitating fair comparison.
When you're working with fractions, sometimes the denominators, or the numbers on the bottom, aren't the same. That's where finding a common denominator comes in - you make the denominators identical so you can start comparing the numerators directly.

Comparing Fractions

Once fractions share a common denominator, comparing them becomes straightforward. The larger the numerator, the larger the fraction. If you picture fractions as pie slices, having a larger numerator means having more slices out of the same-sized pie. This visual can help you intuitively understand why a fraction like \( \frac{42}{96} \) is greater than \( \frac{33}{96} \).
Here's how you can think about it:

• Imagine each fraction as a portion of a pie that has been cut into equal parts based on the denominator.
• The numerators tell you how many parts you get.
• When the parts are equal, the greater numerator means a larger portion of the pie.
  

After the conversion to a common denominator, checking which one is smaller or larger is as simple as comparing their numerators.

Least Common Multiple

Before working with a common denominator, you must find the least common multiple (LCM) of the given denominators. The least common multiple is the smallest number that is a multiple of each of the denominators. For the fractions \(\frac{11}{32}\), \(\frac{15}{48}\), and \(\frac{7}{16}\), you consider 32, 48, and 16. The LCM of these numbers is 96.
Here’s how you figure out the LCM:

• List multiples of each denominator until you find the smallest number common in all lists.
• Work through your multiplication tables to discover shared multiples.
• The smallest shared multiple is your least common multiple - it simplifies subsequent fraction calculations.
  

Identifying the LCM helps in effortlessly transforming all fractions so they can be compared accurately.