Dividing Fractions: Solving 3/4 ÷ 1/2
Dividing 3/4 by 1/2 is simpler than it first appears because dividing by a fraction is the same as multiplying by its reciprocal. The calculation is:
[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} = 1\frac{1}{2} ]
The answer is (1\frac{1}{2}), also written as 3/2 or 1.5. Understanding the operation requires recognizing the meaning of division, changing division into multiplication, and simplifying the resulting fraction correctly.
What Does Dividing 3/4 by 1/2 Mean?
The expression (\frac{3}{4} \div \frac{1}{2}) asks:
How many halves are there in three-fourths?
Division describes equal sharing or grouping. Here's one way to look at it: if a teacher has four equal parts of pizza and gives away one of those parts, the teacher has divided a whole pizza by four. Similarly, dividing (\frac{3}{4}) by (\frac{1}{2}) means determining how many groups of size (\frac{1}{2}) are contained in an amount of (\frac{3}{4}).
This interpretation is especially useful because the divisor is less than one. A fraction such as (\frac{1}{2}) represents a quantity smaller than a whole. This leads to dividing by a number smaller than one generally makes the original quantity larger. Since (\frac{3}{4}) is less than one but greater than (\frac{1}{2}), the answer should be greater than (\frac{3}{4}). The result (1\frac{1}{2}) satisfies that expectation Which is the point..
Real talk — this step gets skipped all the time Not complicated — just consistent..
The Rule for Dividing Fractions
The standard rule for dividing one fraction by another is:
[ \frac{a}{b} \div \frac{c}{d}
\frac{a}{b} \times \frac{d}{c} ]
In other words:
- Keep the first fraction unchanged.
- Change the division sign to multiplication.
- Take the reciprocal of the second fraction.
- Multiply the numerators and denominators.
- Simplify the result if possible.
A reciprocal is the number obtained by switching the numerator and denominator. This leads to the reciprocal of (\frac{1}{2}) is (\frac{2}{1}), which is simply 2. The reciprocal of (\frac{3}{5}), for example, is (\frac{5}{3}).
Applying the rule to the given problem:
[ \frac{3}{4} \div \frac{1}{2}
\frac{3}{4} \times \frac{2}{1} ]
The first fraction stays the same, while the second fraction becomes its reciprocal:
[ \frac{3}{4} \times \frac{2}{1} ]
Now multiply the numerators and denominators:
[ \frac{3 \times 2}{4 \times 1}
\frac{6}{4} ]
Finally, simplify by dividing the numerator and denominator by their greatest common factor, which is 2:
[ \frac{6}{4} \div 2
\frac{3}{2} ]
As a mixed number, (\frac{3}{2}) is written as:
[ 1\frac{1}{2} ]
A Step-by-Step Solution
Here is the complete solution in the most straightforward form:
Step 1: Rewrite the Problem
Start with:
[ \frac{3}{4} \div \frac{1}{2} ]
Step 2: Change Division to Multiplication
The division sign becomes a multiplication sign:
[ \frac{3}{4} \times \frac{1}{2} ]
Step 3: Take the Reciprocal of the Second Fraction
Switch the numerator and denominator of (\frac{1}{2}):
[ \frac{3}{4} \times \frac{2}{1} ]
Step 4: Multiply the Numerators and Denominators
Multiply across:
[ \frac{3 \times 2}{4 \times 1}
\frac{6}{4} ]
Step 5: Simplify
Divide both the numerator and denominator by 2:
[ \frac{6}{4}
\frac{3}{2} ]
Step 6: Choose the Preferred Form
The simplified answer can be written in three equivalent ways:
- Improper fraction: (\frac{3}{2})
- Mixed number: (1\frac{1}{2})
- Decimal: (1.5)
The most appropriate form depends on the instructions or context. For most elementary fraction exercises, (1\frac{1}{2}) is likely the expected answer Most people skip this — try not to. Practical, not theoretical..
Visualizing the Calculation
Fractions can be represented with rectangles, circles, bars, or number lines. Imagine the whole as one rectangle. The amount (\frac{3}{4}) would cover three of the four equal sections:
[ \boxed{\phantom{X}} \quad \boxed{\phantom{X}} \quad \boxed{\phantom{X}} \quad \boxed{\phantom{}} ]
Dividing this amount by (\frac{1}{2}) asks how many half-sized groups are equivalent to the three-fourths amount. Because each half is twice as large as a quarter, dividing by (\frac{1}{2}) is equivalent to doubling (\frac{3}{4}):
[ \frac{3}{4} \times 2 ]
Multiplying (\frac{3}{4}) by 2 gives:
[ \frac{6}{4}
1\frac{1}{2} ]
Another way to see this is to express both fractions in eighths:
[ \frac{3}{4} = \frac{6}{8} \qquad\text{and}\qquad \frac{1}{2} = \frac{4}{8} ]
The question becomes how many groups of 4 eighths are in 6 eighths:
[ 6 \div 4 = \frac{6}{4} = 1\frac{1}{
