Problem 97 - Fundamentals of Mathematics

Question

What part of \(\frac{3}{8}\) is \(\frac{21}{16} ?\)

Step-by-Step Solution

Answer

\( \frac{21}{16} \) is \( \frac{7}{2} \) times \( \frac{3}{8} \).

Step 1: Understand the Question

The question asks us to find out what fraction of \( \frac{3}{8} \) is \( \frac{21}{16} \). This can be rephrased as dividing \( \frac{21}{16} \) by \( \frac{3}{8} \).

Step 2: Set Up the Division

To divide two fractions, we will convert the division into multiplication by the reciprocal. This means we need to multiply \( \frac{21}{16} \) by the reciprocal of \( \frac{3}{8} \), which is \( \frac{8}{3} \). Thus, the operation becomes: \[ \frac{21}{16} \times \frac{8}{3} \]

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together. This gives:\[ \frac{21 \times 8}{16 \times 3} \]

Step 4: Calculate the Numerator and Denominator

Calculate the products:The numerator becomes \( 21 \times 8 = 168 \).The denominator becomes \( 16 \times 3 = 48 \).

Step 5: Simplify the Fraction

The result is \( \frac{168}{48} \). To simplify this fraction, find the greatest common divisor (GCD) of 168 and 48, which is 24. Divide both the numerator and denominator by their GCD:\[ \frac{168 \div 24}{48 \div 24} = \frac{7}{2} \]

Step 6: Express Final Answer

The final simplified fraction after dividing is \( \frac{7}{2} \). This means \( \frac{21}{16} \) is \( \frac{7}{2} \) times \( \frac{3}{8} \).

Key Concepts

Simplifying Fractions

Simplifying a fraction means making it as simple as possible. Just like cleaning up a messy room, simplifying a fraction involves making it tidy by compacting it to its smallest form. The new fraction will still have the same value, just a neater look!

To simplify, you need to find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator evenly without leaving a remainder.

Let's break it down:

• First, list out the factors of both the numerator and the denominator.
• Then, spot the largest factor that both numbers share. That is your GCD!
• Finally, divide both the numerator and the denominator by their GCD.

For example, in simplifying \( \frac{168}{48} \):

• The GCD of 168 and 48 is 24.
• Divide 168 by 24 to get 7.
• Divide 48 by 24 to get 2.

The simplified fraction is \( \frac{7}{2} \). That’s how you make the fraction nice and simple!

Reciprocal of a Fraction

A reciprocal is like a mirror reflection in math. It turns things around by swapping the place of the top number (numerator) and the bottom number (denominator).

The reciprocal of any fraction is found by flipping it. For instance, if you have \( \frac{3}{8} \), its reciprocal is \( \frac{8}{3} \).

Here's where reciprocals come in handy:

• They are essential for turning division of fractions into multiplication.
• This makes the math simpler since multiplying fractions is more straightforward than dividing them.

So, when you have a problem like dividing \( \frac{21}{16} \) by \( \frac{3}{8} \), you change the division into multiplying by the reciprocal. Turn it into \( \frac{21}{16} \times \frac{8}{3} \), and you’re ready to multiply!

Multiplying Fractions

Multiplying fractions is both simple and satisfying. All you need to do is take the numerators of your fractions and multiply them together. Then, do the same with the denominators.

Let's see it in action:

• Start with \( \frac{21}{16} \times \frac{8}{3} \).
• Multiply the numerators:
  • 21 times 8 equals 168.
• Multiply the denominators:
  • 16 times 3 equals 48.

So you get \( \frac{168}{48} \) before any simplification.

Multiplying fractions is one of the most straightforward operations in mathematics. Just remember, it’s all about tops (numerators) together and bottoms (denominators) together. Then, simplify if needed to get the clean, final fraction!