Adding tricky fractions
What is the sum of these fractions?
Age
11 to 14
Challenge level
Problem
Can you find this sum without using a calculator?
$$\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{14}+\frac{1}{15}+\frac{1}{18}+\frac{1}{22}+\frac{1}{24}+\frac{1}{28}+\frac{1}{33}$$
Student Solutions
Answer: 1
Change the order of the fractions in the sum so that fractions whose denominators share more factors are added first: $$\begin{align}&\tfrac{1}{7}+\tfrac{1}{8}+\tfrac{1}{9}+\tfrac{1}{10}+\tfrac{1}{11}+\tfrac{1}{12}+\tfrac{1}{14}+\tfrac{1}{15}+\tfrac{1}{18}+\tfrac{1}{22}+\tfrac{1}{24}+\tfrac{1}{28}+\tfrac{1}{33}\\
&=\hspace{2mm}\tfrac{1}{7}\hspace{1mm}+\tfrac{1}{14}+\tfrac{1}{28}\hspace{2.5mm}+\hspace{3.5mm}\tfrac{1}{8}\hspace{1mm}+\tfrac{1}{12}+\tfrac{1}{24}\hspace{2.5mm}+\hspace{4.5mm}\tfrac{1}{9}\hspace{1mm}+\tfrac{1}{18}\hspace{2.5mm}+\hspace{2.5mm}\tfrac{1}{10}+\tfrac{1}{15}\hspace{2.5mm}+\hspace{2.5mm}\tfrac{1}{11}+\tfrac{1}{22}+\tfrac{1}{33}\\
&=\hspace{1.7mm}\tfrac{4}{28}+\tfrac{2}{28}+\tfrac{1}{28}\hspace{2.2mm}+\hspace{2.5mm}\tfrac{3}{24}+\tfrac{2}{24}+\tfrac{1}{24}\hspace{2.5mm}+\hspace{3.1mm}\tfrac{2}{18}+\tfrac{1}{18}\hspace{2.5mm}+\hspace{2.5mm}\tfrac{3}{30}+\tfrac{2}{30}\hspace{2.5mm}+\hspace{2.5mm}\tfrac{6}{66}+\tfrac{3}{66}+\tfrac{2}{66}\\
&=\hspace{13.2mm}\tfrac{7}{28}\hspace{14mm}+\hspace{14.25mm}\tfrac{6}{24}\hspace{14.25mm}+\hspace{8.5mm}\tfrac{3}{18}\hspace{9mm}+\hspace{8.5mm}\tfrac{5}{30}\hspace{8.5mm}+\hspace{13mm}\tfrac{11}{66}\\
&=\hspace{14.3mm}\tfrac{1}{4}\hspace{14.4mm}+\hspace{15.3mm}\tfrac{1}{4}\hspace{15.3mm}+\hspace{9.4mm}\tfrac{1}{6}\hspace{9.3mm}+\hspace{10.3mm}\tfrac{1}{6}\hspace{9.4mm}+\hspace{13.3mm}\tfrac{1}{6}\\
&=\hspace{3mm}1\end{align}$$