Given the calculation
[tex]6(\frac{1}{2}+\frac{2}{3}+\frac{3}{4})[/tex]1) To solve the calculation with distribution, you have to multiply each term inside the parentheses by 6 and then add the resulting fractions:
[tex]\begin{gathered} 6\cdot\frac{1}{2}+6\cdot\frac{2}{3}+6\cdot\frac{3}{4} \\ \frac{6}{1}\cdot\frac{1}{2}+\frac{6}{1}\cdot\frac{2}{3}+\frac{6}{1}\cdot\frac{3}{4} \\ \frac{6\cdot1}{1\cdot2}+\frac{6\cdot2}{1\cdot3}+\frac{6\cdot3}{1\cdot4} \\ \frac{6}{2}+\frac{12}{3}+\frac{18}{4} \\ 3+4+\frac{9}{2} \end{gathered}[/tex]Now that the multiplications are solved, add the results I will express the fraction 9/2 as a mixed fraction
[tex]\frac{9}{2}\to4\frac{1}{2}[/tex][tex]3+4+4\frac{1}{2}=11\frac{1}{2}[/tex]2) To solve the calculation without distribution you have to solve the addition inside the parentheses first and then multiply the result by 6
[tex]\frac{1}{2}+\frac{2}{3}+\frac{3}{4}[/tex]To add fractions you have to express them using the same denominator, so the first step is to determine the common denominator between 2, 3, and 4.
The least common denominator between these three numbers is 12
2*6=12 → to express 1/2 with denominator 12 you have to multiply the numerator and denominator by 6
[tex]\frac{1\cdot6}{2\cdot6}=\frac{6}{12}[/tex]3*4=12→ to express 2/3 with denominator 12, ou have to multiply the numerator and denominator by 4
[tex]\frac{2\cdot4}{3\cdot4}=\frac{8}{12}[/tex]4*3=12→ to express 3/4 with denominator 12 you have to multiply the numerator and denominator by 3
[tex]\frac{3\cdot3}{4\cdot3}=\frac{9}{12}[/tex]Now you can add the fractions:
[tex]\frac{6}{12}+\frac{8}{12}+\frac{9}{12}=\frac{6+8+9}{12}=\frac{23}{12}[/tex]Next, multiply the result by 6
[tex]6\cdot\frac{23}{12}=\frac{6\cdot23}{1\cdot12}=\frac{138}{12}[/tex]To simplify the fraction you have to divide both values by a common factor, in this case, the common factor between 138 and 12 is 6
[tex]\frac{138\div6}{12\div6}=\frac{23}{2}[/tex]The result can be expressed as a mixed fraction
[tex]\frac{23}{2}=11\frac{1}{2}[/tex]