Use tenths or hundredths to determine the fraction that makes the equation true. Show your thinking.
$\frac{3}{10}+\square=\frac{37}{100}$
The objective is to find the value that fills the blank in the equation \(\frac{3}{10} + \square = \frac{37}{100}\). We let the missing value be denoted by the variable \(x\), so the equation becomes \(\frac{3}{10} + x = \frac{37}{100}\). To solve for \(x\), we need to perform arithmetic operations with fractions, ideally by finding a common denominator to simplify the subtraction.
To add or subtract fractions, they must have the same denominator. Currently, we have one fraction with a denominator of 10 and another with a denominator of 100. We can convert \(\frac{3}{10}\) to an equivalent fraction with a denominator of 100 by multiplying both the numerator and the denominator by 10:
\[\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}.\]
Now, our original equation becomes:
\[\frac{30}{100} + x = \frac{37}{100}.\]
To isolate \(x\), we subtract \(\frac{30}{100}\) from both sides of the equation:
\[x = \frac{37}{100} - \frac{30}{100}.\]
Subtracting the numerators while keeping the denominator constant, we get:
\[x = \frac{37 - 30}{100} = \frac{7}{100}.\]
Thus, the value that makes the equation true is \(\frac{7}{100}\). As a decimal, this is equivalent to \(0.07\).
Understanding common denominators is essential for tasks like adjusting recipes or measuring materials. For example, if you have a recipe that calls for \(\frac{3}{10}\) of a cup of flour and you need to reach a total of \(\frac{37}{100}\) of a cup for a scaled-down batch, finding the difference between these two fractions tells you exactly how much more flour to add. Using the common denominator approach allows you to work accurately with different measurement units.