Fraction Multiplication/Division - ISEE Middle Level Mathematics Achievement

1. Identify the correct operation for: "How many $bc\frac{3}{4}bc$-cup servings are in $bc^2\frac{1}{4}bc$ cups?"

   $2\frac{1}{4} \div \frac{3}{4}$. Determining number of servings in a total amount requires dividing the total by the serving size.

2. A class has $30$ students; $bc\frac{2}{5}bc$ are in band. How many are in band?

   $12$ students. Fraction of students in band is found by multiplying the fraction by total students: $\frac{2}{5} \times 30 = 12$.

3. A tank is $bc\frac{3}{4}bc$ full; you use $bc\frac{2}{5}bc$ of what is in it. What fraction of the tank is used?

   $\frac{3}{10}$. Fraction used of the tank is the product of the fraction full and the fraction used of that: $\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$.

4. What is $bc^2 \times \frac{3}{7}bc$ in simplest form?

   $\frac{6}{7}$. Treat the whole number as a fraction $\frac{2}{1}$ and multiply numerators and denominators: $2 \times 3 = 6$ over $1 \times 7 = 7$.

5. A ribbon is $bc\frac{7}{8}bc$ m long; each piece is $bc\frac{1}{4}bc$ m. How many pieces?

   $\frac{7}{2}$ pieces. Number of pieces is determined by dividing total length by piece length: $\frac{7}{8} \div \frac{1}{4} = \frac{7}{8} \times 4 = \frac{7}{2}$.

6. What is $bc\frac{11}{12} \div \frac{11}{18}bc$ in simplest form?

   $\frac{3}{2}$. Divide by multiplying by reciprocal: $\frac{11}{12} \times \frac{18}{11} = \frac{198}{132}$, simplify by dividing by 66 to get $\frac{3}{2}$.

7. A $12$-inch board is cut to $bc\frac{5}{6}bc$ of its length. What is the new length?

   $10$ inches. Reducing to a fraction of original length requires multiplying: $\frac{5}{6} \times 12 = 10$.

8. What is $bc\frac{3}{4} \times \frac{2}{5}bc$ in simplest form?

   $\frac{3}{10}$. Multiply numerators $3 \times 2 = 6$ and denominators $4 \times 5 = 20$, then simplify $\frac{6}{20}$ by dividing by 2.

9. What is $bc\frac{5}{6}bc$ of $18$?

   $15$. To find a fraction of a whole number, multiply: $\frac{5}{6} \times 18 = 15$, as 18 divided by 6 is 3, times 5 is 15.

10. A runner completes $bc\frac{2}{3}bc$ of a $9$-mile route. How many miles is that?

    $6$ miles. Completing a fraction of a distance involves multiplying the fraction by the total: $\frac{2}{3} \times 9 = 6$.

11. What is $bc\frac{5}{6} \div \frac{2}{3}bc$ in simplest form?

    $\frac{5}{4}$. Divide by multiplying by the reciprocal: $\frac{5}{6} \times \frac{3}{2} = \frac{15}{12}$, then simplify by dividing by 3.

12. A recipe needs $bc\frac{3}{4}bc$ cup sugar; you make $bc\frac{2}{3}bc$ of it. How many cups?

    $\frac{1}{2}$ cup. Scaling a recipe by a fraction requires multiplying the required amount by that fraction: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$.

13. What rule lets you multiply fractions $bc\frac{a}{b} \times \frac{c}{d}bc$ quickly?

    $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. To multiply fractions, multiply the numerators together and the denominators together, resulting in the product of the fractions.

14. What is $bc^2\frac{1}{4} \div \frac{3}{2}bc$ in simplest form?

    $\frac{3}{2}$. Convert mixed number to $\frac{9}{4}$, then divide by multiplying by reciprocal: $\frac{9}{4} \times \frac{2}{3} = \frac{18}{12}$, simplify by 6.

15. You have $bc\frac{3}{5}bc$ lb nuts; servings are $bc\frac{1}{10}bc$ lb. How many servings?

    $6$ servings. Number of servings is found by dividing total amount by serving size: $\frac{3}{5} \div \frac{1}{10} = \frac{3}{5} \times 10 = 6$.

16. What rule lets you divide fractions $bc\frac{a}{b} \div \frac{c}{d}bc$ quickly?

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

17. What is the reciprocal of the fraction $bc\frac{a}{b}bc$ (with $a,b \ne 0$)?

    $\frac{b}{a}$. The reciprocal of a fraction is obtained by swapping its numerator and denominator, ensuring the product with the original is 1.

18. What is $bc^1\frac{1}{2} \times \frac{2}{3}bc$ in simplest form?

    $1$. Convert mixed number to improper fraction $\frac{3}{2}$, then multiply: $\frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1$.

19. What is $bc^3 \div \frac{3}{8}bc$?

    $8$. Divide by multiplying by the reciprocal: $3 \times \frac{8}{3} = \frac{24}{3} = 8$, canceling the 3s.

20. A $2$-liter bottle is poured into cups of $bc\frac{1}{8}bc$ liter. How many cups can be filled?

    $16$ cups. Number of cups is total volume divided by cup size: $2 \div \frac{1}{8} = 2 \times 8 = 16$.

21. A $15$-mile trip is split into equal legs of $bc\frac{3}{4}bc$ mile. How many legs are there?

    $20$ legs. Number of equal legs in a trip is total distance divided by leg length: $15 \div \frac{3}{4} = 15 \times \frac{4}{3} = 20$.

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

    $\frac{1}{6}$. Multiply numerators $7 \times 4 = 28$ and denominators $8 \times 21 = 168$, then simplify by dividing by 28.

23. What is $bc\frac{9}{10} \div \frac{3}{5}bc$ in simplest form?

    $\frac{3}{2}$. Divide by multiplying by the reciprocal: $\frac{9}{10} \times \frac{5}{3} = \frac{45}{30}$, then simplify by dividing by 15.

24. What is $bc\frac{4}{9} \times \frac{27}{8}bc$ in simplest form?

    $\frac{3}{2}$. Multiply numerators $4 \times 27 = 108$ and denominators $9 \times 8 = 72$, then simplify $\frac{108}{72}$ by dividing by 36.

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