Let's solve the problem step-by-step.
We need to find a number such that when [tex]\(\frac{3}{4}\)[/tex] of it is added to [tex]\(3\frac{1}{2}\)[/tex], the result is the same as when [tex]\(\frac{2}{3}\)[/tex] of it is subtracted from [tex]\(6\frac{1}{2}\)[/tex].
1. Let's denote the unknown number by [tex]\(x\)[/tex].
2. First, we convert the mixed numbers to improper fractions:
- [tex]\(3\frac{1}{2} = \frac{7}{2}\)[/tex]
- [tex]\(6\frac{1}{2} = \frac{13}{2}\)[/tex]
3. Write down the expressions:
- [tex]\(\frac{3}{4}x + \frac{7}{2}\)[/tex]
- [tex]\(\frac{13}{2} - \frac{2}{3}x\)[/tex]
4. According to the question, these two expressions are equal:
[tex]\[
\frac{3}{4}x + \frac{7}{2} = \frac{13}{2} - \frac{2}{3}x
\][/tex]
5. To solve for [tex]\(x\)[/tex], we'll eliminate the fractions by finding a common denominator for the terms involving [tex]\(x\)[/tex]. The common denominator for 4 and 3 is 12. So let's rewrite each fraction with this common denominator:
[tex]\[
\left(\frac{3}{4} \times \frac{3}{3}\right)x + \frac{7}{2} = \frac{13}{2} - \left(\frac{2}{3} \times \frac{4}{4}\right)x
\][/tex]
which simplifies to:
[tex]\[
\frac{9}{12}x + \frac{7}{2} = \frac{13}{2} - \frac{8}{12}x
\][/tex]
6. Now, let's combine the terms involving [tex]\(x\)[/tex]:
[tex]\[
\frac{9}{12}x + \frac{8}{12}x = \frac{13}{2} - \frac{7}{2}
\][/tex]
Simplify the left side by adding the fractions:
[tex]\[
\frac{17}{12}x = \frac{13}{2} - \frac{7}{2}
\][/tex]
Simplify the right side by performing the subtraction:
[tex]\[
\frac{17}{12}x = \frac{6}{2} = 3
\][/tex]
7. Finally, solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 \times \frac{12}{17} = \frac{36}{17} \approx 2.1176470588235294
\][/tex]
So, the number [tex]\(x\)[/tex] is approximately [tex]\(2.1176470588235294\)[/tex].
