1. Exact fraction arithmetic

Answer: $\frac{53}{30}$

Write both over the common denominator $30$, combine the numerators, then cancel: $\frac{53}{30}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $\frac{9}{11}$

Answer: $\frac{64}{27}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{16}{9}\right)^{3/2} = \frac{64}{27}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{27}{64}$
• Took the root but never applied the power → $\frac{4}{3}$
• Turned the fraction over and stopped there → $\frac{16}{9}$

Answer: $\frac{13}{4}$

The bars mean the inside is $7$ away from zero in EITHER direction, so there are two equations: $4 x - 6 = 7$ and $4 x - 6 = -7$. They give $\frac{13}{4}$ and $- \frac{1}{4}$; the larger is $\frac{13}{4}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{1}{4}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{13}{4}$

Answer: $\frac{19}{33}$

Multiplication comes before addition, so work out $\dfrac{8}{9} \times \dfrac{6}{11} = \frac{16}{33}$ first, then add $\dfrac{1}{11}$: $\frac{19}{33}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{194}{363}$

Answer: $- \frac{49}{60}$

Write both over the common denominator $60$, combine the numerators, then cancel: $- \frac{49}{60}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $0$

Answer: $\frac{9}{16}$

A negative exponent turns the fraction over, and the fraction $\frac{2}{3}$ means the cube root raised to the power $2$: $\left(\dfrac{27}{64}\right)^{2/3} = \frac{9}{16}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{16}{9}$
• Took the root but never applied the power → $\frac{3}{4}$
• Turned the fraction over and stopped there → $\frac{27}{64}$

Answer: $\frac{7}{5}$

The bars mean the inside is $9$ away from zero in EITHER direction, so there are two equations: $5 x + 2 = 9$ and $5 x + 2 = -9$. They give $\frac{7}{5}$ and $- \frac{11}{5}$; the larger is $\frac{7}{5}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{11}{5}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{7}{5}$

Answer: $\frac{63}{10}$

Multiplication comes before addition, so work out $\dfrac{9}{5} \times \dfrac{5}{2} = \frac{9}{2}$ first, then add $\dfrac{9}{5}$: $\frac{63}{10}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $9$

Answer: $- \frac{77}{30}$

Write both over the common denominator $30$, combine the numerators, then cancel: $- \frac{77}{30}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $-1$

Answer: $\frac{8}{125}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{4}{25}\right)^{3/2} = \frac{8}{125}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{125}{8}$
• Took the root but never applied the power → $\frac{2}{5}$
• Turned the fraction over and stopped there → $\frac{4}{25}$

Answer: $\frac{9}{5}$

The bars mean the inside is $11$ away from zero in EITHER direction, so there are two equations: $5 x + 2 = 11$ and $5 x + 2 = -11$. They give $\frac{9}{5}$ and $- \frac{13}{5}$; the larger is $\frac{9}{5}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{13}{5}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{9}{5}$

Answer: $\frac{351}{392}$

Multiplication comes before addition, so work out $\dfrac{1}{7} \times \dfrac{1}{7} = \frac{1}{49}$ first, then add $\dfrac{7}{8}$: $\frac{351}{392}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{57}{392}$

Answer: $\frac{29}{24}$

Write both over the common denominator $24$, combine the numerators, then cancel: $\frac{29}{24}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $\frac{8}{11}$

Answer: $4$

A negative exponent turns the fraction over, and the fraction $\frac{2}{3}$ means the cube root raised to the power $2$: $\left(\dfrac{8}{1}\right)^{2/3} = 4$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{1}{4}$
• Took the root but never applied the power → $2$
• Turned the fraction over and stopped there → $8$

Answer: $\frac{23}{6}$

The bars mean the inside is $15$ away from zero in EITHER direction, so there are two equations: $6 x - 8 = 15$ and $6 x - 8 = -15$. They give $\frac{23}{6}$ and $- \frac{7}{6}$; the larger is $\frac{23}{6}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{7}{6}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{23}{6}$

Answer: $\frac{57}{88}$

Multiplication comes before addition, so work out $\dfrac{6}{11} \times \dfrac{1}{2} = \frac{3}{11}$ first, then add $\dfrac{3}{8}$: $\frac{57}{88}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{81}{176}$

Answer: $\frac{63}{20}$

Write both over the common denominator $20$, combine the numerators, then cancel: $\frac{63}{20}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $\frac{14}{9}$

Answer: $\frac{25}{4}$

A negative exponent turns the fraction over, and the fraction $\frac{2}{3}$ means the cube root raised to the power $2$: $\left(\dfrac{125}{8}\right)^{2/3} = \frac{25}{4}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{4}{25}$
• Took the root but never applied the power → $\frac{5}{2}$
• Turned the fraction over and stopped there → $\frac{125}{8}$

Answer: $\frac{1}{2}$

The bars mean the inside is $4$ away from zero in EITHER direction, so there are two equations: $4 x + 2 = 4$ and $4 x + 2 = -4$. They give $\frac{1}{2}$ and $- \frac{3}{2}$; the larger is $\frac{1}{2}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{3}{2}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{1}{2}$

Answer: $\frac{241}{360}$

Multiplication comes before addition, so work out $\dfrac{1}{8} \times \dfrac{5}{9} = \frac{5}{72}$ first, then add $\dfrac{3}{5}$: $\frac{241}{360}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{29}{72}$

Answer: $\frac{23}{40}$

Write both over the common denominator $40$, combine the numerators, then cancel: $\frac{23}{40}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $-2$

Answer: $\frac{1000}{27}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{100}{9}\right)^{3/2} = \frac{1000}{27}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{27}{1000}$
• Took the root but never applied the power → $\frac{10}{3}$
• Turned the fraction over and stopped there → $\frac{100}{9}$

Answer: $- \frac{2}{5}$

The bars mean the inside is $2$ away from zero in EITHER direction, so there are two equations: $5 x + 4 = 2$ and $5 x + 4 = -2$. They give $- \frac{2}{5}$ and $- \frac{6}{5}$; the larger is $- \frac{2}{5}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $- \frac{6}{5}$
• Solved the positive branch, then negated the answer instead of solving the second branch → $\frac{2}{5}$

Answer: $\frac{1088}{605}$

Multiplication comes before addition, so work out $\dfrac{4}{11} \times \dfrac{6}{11} = \frac{24}{121}$ first, then add $\dfrac{8}{5}$: $\frac{1088}{605}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{648}{605}$

Answer: $- \frac{39}{110}$

Write both over the common denominator $110$, combine the numerators, then cancel: $- \frac{39}{110}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $4$

Answer: $\frac{27}{8}$

A negative exponent turns the fraction over, and the fraction $\frac{3}{2}$ means the square root raised to the power $3$: $\left(\dfrac{9}{4}\right)^{3/2} = \frac{27}{8}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{8}{27}$
• Took the root but never applied the power → $\frac{3}{2}$
• Turned the fraction over and stopped there → $\frac{9}{4}$

Answer: $\frac{1}{2}$

The bars mean the inside is $9$ away from zero in EITHER direction, so there are two equations: $4 x + 7 = 9$ and $4 x + 7 = -9$. They give $\frac{1}{2}$ and $-4$; the larger is $\frac{1}{2}$.

Common mistakes, and the answer each one gives:

• Gave the smaller of the two solutions → $-4$
• Solved the positive branch, then negated the answer instead of solving the second branch → $- \frac{1}{2}$

Answer: $\frac{94}{77}$

Multiplication comes before addition, so work out $\dfrac{8}{11} \times \dfrac{1}{2} = \frac{4}{11}$ first, then add $\dfrac{6}{7}$: $\frac{94}{77}$.

Common mistakes, and the answer each one gives:

• Worked strictly left to right, adding before multiplying → $\frac{61}{77}$

Answer: $- \frac{62}{55}$

Write both over the common denominator $55$, combine the numerators, then cancel: $- \frac{62}{55}$.

Common mistakes, and the answer each one gives:

• Added (or subtracted) numerators and denominators separately → $- \frac{2}{3}$

Answer: $\frac{25}{9}$

A negative exponent turns the fraction over, and the fraction $\frac{2}{3}$ means the cube root raised to the power $2$: $\left(\dfrac{125}{27}\right)^{2/3} = \frac{25}{9}$.

Common mistakes, and the answer each one gives:

• Ignored the minus sign, so the fraction was never turned over → $\frac{9}{25}$
• Took the root but never applied the power → $\frac{5}{3}$
• Turned the fraction over and stopped there → $\frac{125}{27}$