Problem 48 - Prealgebra

Question

Simplify each expression as much as possible. Find the quotient of \(\frac{4}{5}\) and \(\frac{16}{25}\)

Step-by-Step Solution

Answer

The quotient is \( \frac{5}{4} \).

Step 1: Understand the Operation

To find the quotient of two fractions, you need to divide the first fraction by the second. Dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Reciprocal of the Second Fraction

Find the reciprocal of the second fraction \( \frac{16}{25} \). The reciprocal is found by flipping the numerator and the denominator, resulting in \( \frac{25}{16} \).

Step 3: Multiply the Fractions

Multiply the first fraction \( \frac{4}{5} \) by the reciprocal of the second fraction \( \frac{25}{16} \). The multiplication of fractions involves multiplying their numerators and denominators separately. This gives: \[ \frac{4}{5} \times \frac{25}{16} = \frac{4 \times 25}{5 \times 16} \].

Step 4: Simplify the Expression

Calculate the multiplied expression \( \frac{4 \times 25}{5 \times 16} = \frac{100}{80} \). Simplify the resulting fraction by finding the greatest common divisor of 100 and 80, which is 20. Divide both the numerator and the denominator by 20: \[ \frac{100 \div 20}{80 \div 20} = \frac{5}{4} \].

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key when you're dealing with division involving fractions. A reciprocal is what you get when you flip a fraction upside down. This means you switch the numerator (top number) and the denominator (bottom number). It may sound tricky at first, but it's really quite simple once you get the hang of it! For example, the reciprocal of \( \frac{16}{25} \) is \( \frac{25}{16} \).

Reciprocals are incredibly useful when you have to divide fractions. Instead of actually dividing, you can multiply by the reciprocal. This can make your calculations much easier and faster. Just remember:

• Take the original fraction.
• Swap the numerator and the denominator.
• That's your reciprocal!

Multiplication of Fractions

Once you have the reciprocal, you can convert division into a multiplication operation. To multiply fractions, you multiply the numerators together and the denominators together. This will give you a new fraction, which could be larger or smaller than the original ones. Using our example, to multiply \( \frac{4}{5} \) and the reciprocal \( \frac{25}{16} \), simply multiply the numerators:

• Numerators: \( 4 \times 25 = 100 \)
• Denominators: \( 5 \times 16 = 80 \)

So, the product of these fractions is \( \frac{100}{80} \). Fraction multiplication is straightforward because:

• It does not require a common denominator.
• Just multiply straight across.

Simplifying Fractions

After multiplying fractions, your result can often look more complicated than necessary. That's where simplifying comes in. Simplifying a fraction means making the numbers as small as possible while still keeping them whole numbers. In essence, you are reducing the fraction.

Take our fraction \( \frac{100}{80} \). Both numbers can be divided by their greatest common divisor (GCD). For 100 and 80, the GCD is 20. You split both the numerator and the denominator by this number, and voilà, you get \( \frac{5}{4} \). The steps are:

• Identify the greatest number that divides into both numbers evenly (GCD).
• Divide both the numerator and the denominator by the GCD.
• Write down the simplified fraction.

Always remember, a simplified fraction is easier to work with and often gives you a clearer view of the problem at hand. This can be especially helpful when dealing with more complex calculations or when presenting your answer.