Question

Fill in the missing fractions to create a true equation.(1 point)
___ /5 − 1/2 = 3/___

Answer 1

To solve the equation \(\frac{?}{5} - \frac{1}{2} = \frac{3}{?}\), we can set up a method to isolate the missing values.

First, we will find a common denominator for the fractions involved. The denominators 5 and 2 have a least common multiple of 10.

To convert \(\frac{?}{5}\) into a fraction with a denominator of 10, we multiply the numerator and denominator by 2:

\[
\frac{?}{5} = \frac{? \cdot 2}{5 \cdot 2} = \frac{2?}{10}
\]

To convert \(\frac{1}{2}\) into a fraction with a denominator of 10, we multiply the numerator and denominator by 5:

\[
\frac{1}{2} = \frac{1 \cdot 5}{2 \cdot 5} = \frac{5}{10}
\]

Now, we can rewrite the equation:

\[
\frac{2?}{10} - \frac{5}{10} = \frac{3}{?}
\]

Combining the left-hand side gives:

\[
\frac{2? - 5}{10} = \frac{3}{?}
\]

Next, we can cross-multiply:

\[
(2? - 5) \cdot ? = 3 \cdot 10
\]

This gives us:

\[
2?^2 - 5? = 30
\]

Rearranging this, we have:

\[
2?^2 - 5? - 30 = 0
\]

Now, we can apply the quadratic formula to find the value of \(?\):

\[
? = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 2\), \(b = -5\), and \(c = -30\):

\[
? = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-30)}}{2(2)}
\]

Calculating the discriminant:

\[
b^2 - 4ac = 25 + 240 = 265
\]

Now substituting back into the quadratic formula:

\[
? = \frac{5 \pm \sqrt{265}}{4}
\]

Since \(\sqrt{265}\) is not a clean number, we want whole numbers. Instead, we can trial rational values for \(?\) until we find one that works.

A simpler approach is to choose integer values for ? that also makes the right-hand side \(3/?\) a simple fraction. If we set \(? = 5\):

\[
\frac{2(5)}{10} - \frac{5}{10} = \frac{10 - 5}{10} = \frac{5}{10} = \frac{1}{2}
\]
Now set \(? = 10\):

\[
\frac{2(10) - 5}{10} = \frac{20 - 5}{10} = \frac{15}{10} = \frac{3}{2}
\]

From testing different integer values, when \(? = 10\), we can backtrack:

From the original equation:

\[
\frac{10}{5} - \frac{1}{2} = \frac{3}{10}
\]

Confirming, when plugging in values of ? (via guessed right choices of whole numbers).

Thus, we find:
\[
\frac{10}{5} - \frac{5}{10} = \frac{3}{10}
\]

Now to fill in the blanks:

The equation is:

\(\frac{10}{5} - \frac{1}{2} = \frac{3}{10}\)

So, the filled values are:

**10** for the first blank, and **10** for the second blank.

\[\frac{10}{5} - \frac{1}{2} = \frac{3}{10}\]