The phrase 3 4 divided by 2 5 is a shorthand way of expressing the division of the fraction three‑fourths by the fraction two‑fifths. In mathematical notation this operation is written as
[ \frac{3}{4} \div \frac{2}{5} ]
and it appears frequently in algebra, arithmetic, and real‑world problem solving. Understanding how to evaluate this expression not only gives you a single numeric answer but also reinforces the underlying principles of fraction arithmetic, reciprocal relationships, and simplification techniques Most people skip this — try not to. And it works..
Understanding the Operation
Before diving into a step‑by‑step solution, it helps to clarify what “dividing by a fraction” actually means. ” In the case of fractions, this question can be answered by converting the division into a multiplication problem using the reciprocal of the divisor. When you divide one quantity by another, you are essentially asking, “How many times does the divisor fit into the dividend?The reciprocal of a fraction is obtained by swapping its numerator and denominator; for example, the reciprocal of (\frac{2}{5}) is (\frac{5}{2}).
Step‑by‑Step Solution
Step 1: Write the division as a fraction
Begin by expressing the division in a clear format:
[ \frac{3}{4} \div \frac{2}{5} ]
This representation makes it easier to apply the reciprocal rule in the next step Small thing, real impact..
Step 2: Find the reciprocal of the divisor
Identify the divisor, which is (\frac{2}{5}), and compute its reciprocal:
[ \text{Reciprocal of } \frac{2}{5} = \frac{5}{2} ]
Step 3: Multiply the dividend by the reciprocal
Replace the division sign with a multiplication sign and multiply the dividend (\frac{3}{4}) by the reciprocal \
Step 4: Perform the multiplication
Now replace the division symbol with a multiplication sign and multiply the numerators together and the denominators together:
[ \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}. ]
Step 5: Simplify the result
The fraction (\frac{15}{8}) is already in lowest terms because 15 and 8 share no common divisor other than 1. If a mixed number is preferred, convert it:
[ \frac{15}{8}=1\frac{7}{8}. ]
Thus, the exact value of (\displaystyle \frac{3}{4} \div \frac{2}{5}) is (\frac{15}{8}) (or (1\frac{7}{8})).
Step 6: Verify the answer (optional)
To double‑check, multiply the result by the original divisor:
[ \frac{15}{8} \times \frac{2}{5} = \frac{30}{40} = \frac{3}{4}, ]
which indeed reproduces the dividend, confirming the calculation is correct Worth knowing..
Real‑World Context
This type of division often arises when scaling recipes, converting units, or determining rates. Here's a good example: if a recipe calls for (\frac{3}{4}) cup of sugar but you need to know how many (\frac{2}{5})‑cup servings fit into that amount, the answer (\frac{15}{8}) tells you there are (1\frac{7}{8}) such servings Worth keeping that in mind. Turns out it matters..
Conclusion
Dividing fractions may seem intimidating at first, but by converting the division into multiplication with the reciprocal, the process becomes straightforward. Applying this technique to (\frac{3}{4} \div \frac{2}{5}) yields the simplified result (\frac{15}{8}) (or (1\frac{7}{8})). Mastering this method not only provides quick solutions to arithmetic problems but also strengthens the conceptual understanding of how fractions interact, a skill that proves invaluable across mathematics and everyday applications Worth keeping that in mind..
Additional Tips and Common Mistakes to Avoid
Do not change the first fraction
A common error is to find the reciprocal of the dividend instead of the divisor. In a division problem, only the second fraction—the divisor—should be flipped.
For example:
[ \frac{5}{6} \div \frac{1}{3} ]
should become
[ \frac{5}{6} \times \frac{3}{1} ]
not
[ \frac{6}{5} \times \frac{3}{1}. ]
The divisor is always the fraction being divided by.
Cross-cancel before multiplying
Cross-canceling can make multiplication easier, especially when the numbers are large. For example:
[ \frac{4}{7} \div \frac{6}{5}
\frac{4}{7} \times \frac{5}{6} ]
Before multiplying, notice that 4 and 6 share a common factor of 2:
[ \frac{4}{7} \times \frac{5}{6}
\frac{2}{7} \times \frac{5}{3}
\frac{10}{21}. ]
This avoids unnecessary simplifying at the end The details matter here..
Convert whole numbers and mixed numbers first
Whole numbers can be written as fractions by placing them over 1:
[ 2 = \frac{2}{1}. ]
Mixed numbers should be converted to improper fractions before dividing:
[ 1\frac{1}{2} = \frac{3}{2}. ]
For example:
[ 1\frac{1}{2} \div \frac{1}{4}
\frac{3}{2} \div \frac{1}{4}
\frac{3}{2} \times 4
]
Remember that division by zero is undefined
Just as with whole numbers, dividing by zero is not allowed. A fraction such as (\frac{0}{5}) is equal to 0, so it cannot be used as a divisor:
[ \frac{3}{4} \div \frac{0}{5} ]
is undefined.
Quick Practice Examples
[ \frac{2}{3} \div \frac{4}{5}
\frac{2}{3} \times \frac{5}{4}
\frac{10}{12}
\frac{5}{6} ]
[ \frac{7}{8} \div 3
\frac{7}{8
