Fraction Operations - ISEE Upper Level Mathematics Achievement

1. What is the sign rule for a negative fraction $\frac{-a}{b}$ compared to $\frac{a}{-b}$?

   $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$. A negative sign can be placed in the numerator, denominator, or in front of the fraction equivalently.

2. What is the general method to subtract $\frac{a}{b} - \frac{c}{d}$ using a common denominator?

   $\frac{ad-bc}{bd}$. Convert to a common denominator of $bd$ and subtract the adjusted numerators.

3. What is $\left(-\frac{2}{3}\right) + \frac{5}{6}$ in simplest form?

   $\frac{1}{6}$. Convert to a common denominator of 6 and add the numerators.

4. What does it mean for $\frac{a}{b}$ to be in simplest form?

   $\gcd(a,b)=1$. The fraction is in simplest form when the numerator and denominator have no common factors other than 1.

5. What is the rule for multiplying fractions, $\frac{a}{b} \times \frac{c}{d}$?

   $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. Multiply the numerators together and the denominators together.

6. What is the least common denominator (LCD) of denominators $b$ and $d$?

   $\mathrm{LCD} = \mathrm{lcm}(b,d)$. The least common denominator is the least common multiple of the denominators.

7. What is the reciprocal of a nonzero fraction $\frac{a}{b}$?

   $\frac{b}{a}$. The reciprocal is obtained by swapping the numerator and denominator.

8. What is $\frac{1}{2} + \frac{2}{5}$ in simplest form?

   $\frac{9}{10}$. Convert to a common denominator of 10 and add the numerators.

9. What is $\frac{7}{9} \div \frac{14}{3}$ in simplest form?

   $\frac{1}{6}$. Multiply by the reciprocal, then simplify by dividing by 21.

10. What is $\left(-\frac{4}{7}\right) \div \left(\frac{2}{3}\right)$ in simplest form?

    $-\frac{6}{7}$. Multiply by the reciprocal, retaining the negative sign, and simplify by dividing by 2.

11. What is $\left(-\frac{3}{5}\right) \times \left(\frac{10}{9}\right)$ in simplest form?

    $-\frac{2}{3}$. Multiply numerators and denominators, retaining the negative sign, and simplify by dividing by 15.

12. What is $\frac{3}{4} - \frac{2}{3}$ in simplest form?

    $\frac{1}{12}$. Convert to a common denominator of 12 and subtract the numerators.

13. What is $\frac{4}{7} \times \frac{21}{8}$ in simplest form?

    $\frac{3}{2}$. Multiply numerators and denominators, then simplify by dividing by 28.

14. What is $\frac{2}{3} + \frac{1}{6}$?

    $\frac{5}{6}$. Convert to a common denominator of 6 and add the numerators.

15. What is $\frac{11}{12} - \frac{5}{18}$ in simplest form?

    $\frac{23}{36}$. Convert to a common denominator of 36 and subtract the numerators; the result is already simplest.

16. What is $\frac{5}{12} - \frac{1}{4}$?

    $\frac{1}{6}$. Convert to a common denominator of 12, subtract the numerators, and simplify by dividing by 2.

17. What is $\frac{3}{5} \times \frac{10}{9}$ in simplest form?

    $\frac{2}{3}$. Multiply numerators and denominators, then simplify by dividing by 15.

18. What is $\frac{3}{8} + \frac{1}{8}$?

    $\frac{1}{2}$. Add the numerators over the common denominator and simplify by dividing by 4.

19. What is the rule for subtracting fractions with the same denominator, $\frac{a}{b} - \frac{c}{b}$?

    $\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}$. Subtract the numerators while keeping the common denominator.

20. What is $\frac{2}{9} + \frac{5}{12}$ in simplest form?

    $\frac{23}{36}$. Convert to a common denominator of 36 and add the numerators; the result is already simplest.

21. What is the rule for adding fractions with the same denominator, $ \frac{a}{b} + \frac{c}{b}$?

    $\frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}$. Add the numerators while keeping the common denominator.

22. What is the general method to add $\frac{a}{b} + \frac{c}{d}$ using a common denominator?

    $\frac{ad+bc}{bd}$. Convert to a common denominator of $bd$ and add the adjusted numerators.

23. What is $\frac{5}{6} \div \frac{10}{9}$ in simplest form?

    $\frac{3}{4}$. Multiply by the reciprocal, then simplify by dividing by 15.

24. What is $\frac{7}{10} - \frac{3}{10}$?

    $\frac{2}{5}$. Subtract the numerators over the common denominator and simplify by dividing by 2.

25. What is the rule for dividing fractions, $\frac{a}{b} \div \frac{c}{d}$, where $c \neq 0$?

    $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. Dividing by a fraction is equivalent to multiplying by its reciprocal.

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