Question

$$ \begin{aligned} &\frac{3}{54}-\frac{3}{56}\\\ &\begin{array}{|l|l|} \hline \text { Select... } & \nabla \\ \hline> \\ \hline< \\ \hline= \\ \hline \end{array} \end{aligned} $$

Short answer

\( > \)

Step-by-step solution

Simplify each fraction

Simplify the first fraction, \(\frac{3}{54}\). Both 3 and 54 are divisible by 3: \(\frac{3 ÷ 3}{54 ÷ 3} = \frac{1}{18}\). Next, simplify the second fraction, \(\frac{3}{56}\). Both 3 and 56 have no common factors other than 1, so \(\frac{3}{56}\) remains the same.

Compare the simplified fractions

To compare \(\frac{1}{18}\) and \(\frac{3}{56}\), find a common denominator or convert to decimal form. Converting to decimals, \(\frac{1}{18} \approx 0.056\) and \(\frac{3}{56} \approx 0.05357\).

Determine the relationship

Comparing 0.056 and 0.05357, we see that 0.056 is greater than 0.05357.

Select the appropriate symbol

Since \(\frac{1}{18} > \frac{3}{56}\), the correct symbol is \( > \).

Key concepts

fraction simplification

Fraction simplification makes fractions easier to work with. To simplify a fraction, both the numerator (top number) and the denominator (bottom number) should be divided by their greatest common divisor (GCD). For example, to simplify \(\frac{3}{54}\), find the GCD of 3 and 54, which is 3. Dividing both by 3, we get \(\frac{1}{18}\). Not all fractions can be simplified. In some cases, like \(\frac{3}{56}\), the GCD is 1, and the fraction remains unchanged. Simplification helps us in comparing fractions more easily.

decimal conversion

Converting fractions to decimals can help in comparison. To convert, divide the numerator by the denominator. For \(\frac{1}{18}\), the result is approximately 0.056. For \(\frac{3}{56}\), it’s around 0.05357. Decimal conversion shows which fraction is greater or smaller at a glance. In our example, since 0.056 > 0.05357, \(\frac{1}{18}\) is greater than \(\frac{3}{56}\).

common denominator

Finding a common denominator is another method for comparing fractions. Both fractions are converted to have the same denominator. First, determine the least common multiple (LCM) of the denominators. For 18 and 56, the LCM is 504. Convert each fraction: \(\frac{1}{18}\) becomes \(\frac{28}{504}\), and \(\frac{3}{56}\) becomes \(\frac{27}{504}\). Comparing \(\frac{28}{504}\) and \(\frac{27}{504}\) is straightforward; \(\frac{28}{504}\) is greater. So, \(\frac{1}{18}\) > \(\frac{3}{56}\).