Problem 102
Question
Divide. $$\frac{7}{8} \div \frac{7}{16}$$
Step-by-Step Solution
Verified Answer
The result of the division is 2.
1Step 1: Understand Division of Fractions
When dividing fractions, we multiply by the reciprocal. The reciprocal of a fraction is obtained by flipping its numerator and denominator.
2Step 2: Determine the Reciprocal of the Second Fraction
The given problem is \( \frac{7}{8} \div \frac{7}{16} \). For division, we take the reciprocal of \( \frac{7}{16} \), resulting in \( \frac{16}{7} \).
3Step 3: Multiply the First Fraction by the Reciprocal
Replace the division sign with multiplication, and use the reciprocal of \( \frac{7}{16} \). The problem becomes \( \frac{7}{8} \times \frac{16}{7} \).
4Step 4: Simplify the Multiplication
Multiply the numerators together and the denominators together separately: \( 7 \times 16 = 112 \) and \( 8 \times 7 = 56 \). This gives us \( \frac{112}{56} \).
5Step 5: Simplify the Fraction
Simplify \( \frac{112}{56} \). Both numerator and denominator are divisible by 56. Dividing both, we get \( \frac{112}{56} = 2 \).
Key Concepts
Reciprocal of a FractionSimplifying FractionsFraction Multiplication
Reciprocal of a Fraction
Understanding the reciprocal of a fraction is key when dividing fractions. A reciprocal flips the numerator and the denominator of the fraction. For example, if you have the fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \).
This concept is useful because dividing by a fraction is the same as multiplying by its reciprocal. In simple terms, dividing \( \frac{7}{8} \) by \( \frac{7}{16} \) is like asking, "How many times does \( \frac{7}{16} \) fit into \( \frac{7}{8} \)?" By using the reciprocal, you transform the division problem into a multiplication problem, which is often easier to solve.
This concept is useful because dividing by a fraction is the same as multiplying by its reciprocal. In simple terms, dividing \( \frac{7}{8} \) by \( \frac{7}{16} \) is like asking, "How many times does \( \frac{7}{16} \) fit into \( \frac{7}{8} \)?" By using the reciprocal, you transform the division problem into a multiplication problem, which is often easier to solve.
Simplifying Fractions
Simplifying fractions is the process of making the fraction as simple as possible by ensuring that the numerator and denominator are as small as possible while retaining the value. To simplify, you need to find the greatest common divisor (GCD) of both the numerator and denominator and divide them by this number.
Let's look at an example: When we multiply the fractions \( \frac{7}{8} \times \frac{16}{7} \), we end up with \( \frac{112}{56} \). Here, both 112 and 56 are divisible by 56. Dividing gives \( \frac{112}{56} = 2 \).
Always check if the numerator and denominator can be divided by the same number until you cannot simplify any further. Simplifying fractions can help make calculations faster and answers clearer.
Let's look at an example: When we multiply the fractions \( \frac{7}{8} \times \frac{16}{7} \), we end up with \( \frac{112}{56} \). Here, both 112 and 56 are divisible by 56. Dividing gives \( \frac{112}{56} = 2 \).
Always check if the numerator and denominator can be divided by the same number until you cannot simplify any further. Simplifying fractions can help make calculations faster and answers clearer.
Fraction Multiplication
When you multiply fractions, you multiply the numerators together and the denominators together to get your answer. This fundamental arithmetic rule is straightforward: For fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \times c}{b \times d} \).
In the division problem \( \frac{7}{8} \div \frac{7}{16} \), after changing the problem to a multiplication one using the reciprocal, we compute \( \frac{7}{8} \times \frac{16}{7} \). The numerators, 7 and 16, are multiplied to give 112; the denominators, 8 and 7, give 56, resulting in \( \frac{112}{56} \).
This approach converts what might seem like a complicated operation into something straightforward. It's one of the reasons multiplication is a favored operation in fraction problems!
In the division problem \( \frac{7}{8} \div \frac{7}{16} \), after changing the problem to a multiplication one using the reciprocal, we compute \( \frac{7}{8} \times \frac{16}{7} \). The numerators, 7 and 16, are multiplied to give 112; the denominators, 8 and 7, give 56, resulting in \( \frac{112}{56} \).
This approach converts what might seem like a complicated operation into something straightforward. It's one of the reasons multiplication is a favored operation in fraction problems!