Closer to one

Here are two sums: A=1/2 + 1/3 and B=1/5 + 1/3.
Which of the two sums is closer in value to 1?
You must show your work and state clearly whether the answer is A or B.

Correct answer:

x = A

Step-by-step explanation:

$$\begin{gathered} A=\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{3+2}{6}=\frac{5}{6}\doteq 0.8333 \\ B=\frac{1}{5}+\frac{1}{3}=\frac{3}{15}+\frac{5}{15}=\frac{3+5}{15}=\frac{8}{15}\doteq 0.5333 \\ {d}_1=1-A=1-\frac{5}{6}=\frac{1\cdot 6}{6}-\frac{5}{6}=\frac{6}{6}-\frac{5}{6}=\frac{6-5}{6}=\frac{1}{6}\doteq 0.1667 \\ {d}_2=1-B=1-\frac{8}{15}=\frac{1\cdot 15}{15}-\frac{8}{15}=\frac{15}{15}-\frac{8}{15}=\frac{15-8}{15}=\frac{7}{15}\doteq 0.4667 \\ |d_1|<|d_2| \\ x=A \end{gathered}$$

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