Multiply Fractions by Whole Numbers - 4th Grade Math

1. Sofia pours $\frac{4}{7}$ liter of water into each bottle. She fills 3 bottles. What is $3 \times \left(\frac{4}{7}\right)$? Use $n \times \left(\frac{a}{b}\right) = \frac{n \times a}{b}$.
   Answer: $\frac{12}{7}$ The correct answer is 12/7, because 3 times 4/7 equals (3 times 4)/7, which is 12/7. Choice A, 12/21, mistakenly multiplies both the numerator and denominator by 3 instead of just the numerator. Choice C, 4/21, multiplies the denominator by 3 but leaves the numerator too small, losing track of all three bottles. Choice D, 7/7, comes from adding the numerator and denominator instead of multiplying by 3.
2. Maya has $3$ groups of $\tfrac{4}{11}$. What is the total?
   Answer: $\tfrac{12}{11}$ The correct answer is D, 12/11, because 3 times 4/11 equals 12/11. Choice A, 12/33, comes from also multiplying the denominator by 3 instead of keeping it the same. Choice B, 4/33, comes from multiplying the denominators together instead of multiplying the numerator by 3. Choice C, 7/11, comes from adding 3 and 4/11 instead of multiplying.
3. Use the fact that $9 \times (4/7) = (9 \times 4) \times (1/7)$ to find the product.
   Answer: $36/7$ Since 9 x 4 = 36, the product 9 x (4/7) equals 36/7. Choice A multiplies the denominators together instead of scaling only the numerator. Choice C multiplies both the numerator and denominator by 9, which is not how whole-number-by-fraction multiplication works. Choice D reflects an addition-based error rather than multiplication.
4. Emma colored $\frac{2}{9}$ of a page. Her friend colored 4 times as much as Emma colored. Together, how much of the page did they color?
   Answer: $\frac{10}{9}$ of the page was colored Emma colored $\frac{2}{9}$. Her friend colored $4 \times \frac{2}{9} = \frac{8}{9}$. Together: $\frac{2}{9} + \frac{8}{9} = \frac{10}{9}$. Choice A only counts Emma's portion plus $\frac{4}{9}$. Choice B is just the friend's portion. Choice D is $6 \times \frac{2}{9}$.
5. Study the equation: $4 \times \frac{a}{5} = \frac{12}{5}$. What value of $a$ makes this equation true?
   Answer: $a = 3$ $a = 3$ is correct because $4 \times \frac{3}{5} = \frac{12}{5}$. $a = 8$ is incorrect because $4 \times \frac{8}{5}$ does not equal $\frac{12}{5}$. $a = 12$ is incorrect because it skips dividing by 4. $a = 48$ is incorrect because it multiplies instead of dividing to isolate $a$.
6. Multiply $7 \times \tfrac{1}{10}$. Remember that $\tfrac{1}{10}$ means one of ten equal parts of a whole.
   Answer: $\tfrac{7}{10}$ Multiplying a whole number by a unit fraction means taking that many copies of the unit fraction: $7 \times \tfrac{1}{10} = \tfrac{7}{10}$. Choice A incorrectly multiplies both the numerator and denominator by 7. Choice B incorrectly multiplies only the denominator by 7. Choice D adds 1 to the numerator instead of multiplying by 7.
7. What is $5 \times \tfrac{2}{3}$?
   Answer: $10/3$ The correct answer is A, 10/3, because 5 times 2/3 equals 10/3. Choice B, 2/15, comes from multiplying the denominators together instead of multiplying the numerator by 5. Choice C, 10/15, comes from also multiplying the denominator by 5 instead of keeping it the same. Choice D, 7/3, comes from adding 5 and 2/3 instead of multiplying.
8. Carlos walks $9$ laps. Each lap is $\tfrac{2}{5}$ mile. What is the total distance Carlos walks?
   Answer: $\tfrac{18}{5}$ The correct answer is A, 18/5 miles, because 9 times 2/5 equals 18/5. Choice B, 2/45, comes from multiplying the denominators together instead of multiplying the numerator by 9. Choice C, 11/5, comes from adding 9 and 2/5 instead of multiplying. Choice D, 18/45, comes from also multiplying the denominator by 9 instead of keeping it the same.
9. Carlos ran $\frac{5}{8}$ of a mile each day for 3 days. Maya ran $\frac{3}{8}$ of a mile each day for 5 days. Who ran farther?
   Answer: They ran the same distance They ran the same distance because Carlos ran $3 \times \frac{5}{8} = \frac{15}{8}$ miles and Maya ran $5 \times \frac{3}{8} = \frac{15}{8}$ miles. Carlos ran farther is incorrect because the two totals are equal. Maya ran farther is incorrect for the same reason. Not enough information is incorrect because both totals can be calculated from the given facts.
10. Jamal walks $\frac{5}{12}$ mile each lap. He walks 6 laps. What is $6 \times \left(\frac{5}{12}\right)$? Think of $\frac{5}{12} = 5 \times \left(\frac{1}{12}\right)$.
    Answer: $\frac{30}{12}$ Multiplying 6 by 5/12 means combining six groups of five twelfths, which gives 30/12, making Choice A correct. Choice B, 5/72, comes from multiplying both the numerator and denominator by 6 instead of only the numerator. Choice C, 30/72, comes from multiplying the denominator by 6 as well, which is not how multiplying a whole number by a fraction works. Choice D, 11/12, comes from adding 6 and 5 instead of multiplying them.
11. What is $5 \times \left(\frac{3}{10}\right)$ written as a fraction?
    Answer: $\frac{15}{10}$ Multiplying a whole number by a fraction scales the numerator: 5 x 3 = 15, so 5 x (3/10) = 15/10. Choice A multiplies both the numerator and denominator by 5 instead of only the numerator. Choice B multiplies the denominator instead of the numerator. Choice D reflects an addition-based error rather than multiplication.
12. Keisha fills 5 small jars. Each jar holds $\tfrac{2}{7}$ cup of beads. What is the total amount of beads, in cups?
    Answer: $\tfrac{10}{7}$ Five jars holding 2/7 cup each gives 5 x (2/7) = 10/7 cups total. Choice A swaps the numerator and denominator, giving an incorrect fraction. Choice C multiplies both the numerator and denominator by 5 instead of only the numerator, so it does not represent multiplication correctly. Choice D only multiplies the denominator, which shrinks the fraction instead of scaling it up.
13. Multiply $2 \times (7/9)$. Use the idea $7/9 = 7 \times (1/9)$, so the product is a multiple of $1/9$.
    Answer: $14/9$ This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. This works because 7/9 is really 7 copies of the unit fraction 1/9, so 2 groups of (7 copies of 1/9) equals (2 × 7) copies of 1/9, which equals 14/9. To multiply 2 × (7/9), we multiply 2 × 7 = 14, and keep the denominator 9, giving 14/9. We can also see this as 2 groups of 7 unit fractions (1/9), which equals 14 unit fractions = 14 × (1/9) = 14/9. Choice C is correct because multiplying: 2 × 7 = 14, keeping denominator 9: 14/9. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 2. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 7/9 is '7 copies of 1/9,' so 2 groups of that is (2 × 7) copies of 1/9 = 14 × (1/9) = 14/9. Connect to repeated addition: 2 × (7/9) = 7/9 + 7/9 = 14/9. Convert improper fractions to mixed numbers for interpretation: 14/9 = 1 5/9.
14. A whole shape is divided into sixths. If 7 of these one-sixth pieces are shaded across one or more wholes, what multiplication expression represents the total shaded amount?
    Answer: $7 \times \frac{1}{6}$ 7 shaded one-sixth pieces means the total is 7 groups of 1/6, or 7 × 1/6. Choice B swaps the roles of the piece count and the piece size. Choice C uses 8 instead of the actual count of 7 shaded pieces. Choice D also swaps which number represents the pieces and which represents the size of each piece.
15. A recipe calls for $\frac{2}{5}$ cup of flour. Jake wants to make 3 batches of the recipe, but he only has a $\frac{1}{5}$ cup measuring scoop. How many scoops of flour will Jake need?
    Answer: 6 scoops of flour total Jake needs $3 \times \frac{2}{5} = \frac{6}{5}$ cups total. Since each scoop is $\frac{1}{5}$ cup, he needs $\frac{6}{5} \div \frac{1}{5} = 6$ scoops. Choice A uses $\frac{2}{5} \times 5 = 2$ batches plus 3 extra. Choice C incorrectly adds $3 + 2 + 3 = 8$. Choice D uses $2 \times 5 = 10$.
16. Multiply $7 \times \left(\frac{2}{9}\right)$. Show you understand that $\frac{2}{9} = 2 \times \left(\frac{1}{9}\right)$, so the total is $(7 \times 2)$ ninths.
    Answer: $\frac{14}{9}$ Multiplying a whole number by a fraction means multiplying the whole number by the numerator: $7 \times 2 = 14$, keeping the denominator 9, giving $\frac{14}{9}$. Choice A comes from multiplying the denominators together instead of the numerator by the whole number. Choice B multiplies both the numerator and denominator by 7, which changes the value unnecessarily. Choice D comes from adding 7 and 2 to get 9 and mistakenly using that as both parts of the fraction. Choice C correctly applies the rule for multiplying a whole number by a fraction.
17. Keisha makes 6 batches of trail mix. Each batch uses $3/10$ cup of raisins. What is $6 \times (3/10)$ cups in all? Use $6 \times (a/b) = (6 \times a)/b$.
    Answer: $18/10$ The correct answer is 18/10, because 6 times 3/10 equals (6 times 3)/10, which is 18/10. Choice B, 3/60, mistakenly multiplies the denominator by 6 instead of the numerator. Choice C, 18/60, multiplies both the numerator and denominator by 6 instead of just the numerator. Choice D, 9/10, comes from adding 6 and 3 instead of multiplying them.
18. Maria has $\frac{3}{8}$ of a pizza. She wants to make 4 lunch portions, each $\frac{3}{8}$ of a pizza. How much pizza will Maria need in total?
    Answer: $\frac{12}{8}$ of a pizza Maria needs 12/8 of a pizza because 4 portions of 3/8 each means 4 times 3/8, which equals 12/8. Choice A adds 3/8 + 4/8 rather than multiplying. Choice C incorrectly multiplies the denominators together for a nonsensical result. Choice D multiplies both the numerator and denominator by 4, which doesn't represent repeated addition correctly.
19. What is $8 \times \frac{5}{12}$?
    Answer: $40/12$ Multiplying a whole number by a fraction means multiplying the whole number by the numerator and keeping the denominator: $8 \times 5 = 40$, so the product is $40/12$. Choice A comes from adding 8 and 5 instead of multiplying them. Choice B mistakenly multiplies the denominators together instead of keeping the original denominator. Choice D multiplies both the numerator and denominator by 8, which changes the value unnecessarily. Choice C correctly applies the rule for multiplying a whole number by a fraction.
20. What is $7 \times \frac{1}{4}$?
    Answer: $7/4$ Multiplying a whole number by a fraction means multiplying the whole number by the numerator and keeping the denominator: $7 \times 1 = 7$, so the product is $7/4$. Choice B multiplies both the numerator and the denominator by 7, which changes the value unnecessarily. Choice C uses only the denominator multiplication without adjusting the numerator, giving a much smaller value. Choice D comes from adding 7 and 1 instead of multiplying them. Choice A correctly applies the rule for multiplying a whole number by a fraction.

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