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 $$2 \times 5 = 10$$.

Carlos: $$3 \times \frac{5}{8} = \frac{15}{8}$$ miles. Maya: $$5 \times \frac{3}{8} = \frac{15}{8}$$ miles. They ran equal distances. Choice A assumes Carlos ran more. Choice B assumes Maya ran more. Choice D incorrectly calculates the difference as $$\frac{8}{8}$$.

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: n × (a/b) = (n × a)/b. This works because a/b is really a copies of the unit fraction 1/b, so n groups of (a copies of 1/b) equals (n × a) copies of 1/b, which equals (n × a)/b. For example, 3 × (2/5) means 3 groups of two-fifths, which equals 6 fifths total, or 6/5. To multiply 3 × (2/5), we multiply 3 × 2 = 6, and keep the denominator 5, giving 6/5. We can also see this as 3 groups of 2 unit fractions (1/5), which equals 6 unit fractions = 6 × (1/5) = 6/5. Choice B is correct because multiplying: 3 × 2 = 6, keeping denominator 5: 6/5; using unit fractions: 3 × (2 × (1/5)) = (3 × 2) × (1/5) = 6 × (1/5) = 6/5; repeated addition: 2/5 + 2/5 + 2/5 = 6/5. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents multiplying both numerator and denominator by 3, giving 6/15, which happens when students incorrectly think both parts multiply by n. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: a/b is 'a copies of 1/b,' so n groups of that is (n × a) copies of 1/b = (n × a)/b. Use visuals: draw 3 groups of 2/5 (three sets of 2 shaded fifths), count total fifths: 6 fifths = 6/5. Practice with unit fractions first: 3 × (1/5) = 3/5 (easier to see), then extend: 3 × (2/5) = 3 × [2 × (1/5)] = 6 × (1/5) = 6/5. Connect to repeated addition: 3 × (2/5) = 2/5 + 2/5 + 2/5 = 6/5. Convert improper fractions to mixed numbers for interpretation: 6/5 = 1 1/5. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication, and forgetting that denominator stays the same.

$$6 \times \frac{3}{4} = \frac{18}{4} = 18 \times \frac{1}{4}$$. This can be written as $$6 \times 3 \times \frac{1}{4}$$ or $$3 \times 6 \times \frac{1}{4}$$ due to the commutative property. All expressions equal $$\frac{18}{4}$$ cups.

Maria needs 4 portions, each the size of $$\frac{3}{8}$$. This means $$4 \times \frac{3}{8} = \frac{4 \times 3}{8} = \frac{12}{8}$$. Choice A incorrectly adds $$4 + \frac{3}{8}$$. Choice C incorrectly computes $$\frac{4}{8+3}$$. Choice D multiplies both numerator and denominator by 4.

Total needed: $$8 \times \frac{2}{3} = \frac{16}{3}$$ yards. She has $$\frac{10}{3}$$ yards. Additional needed: $$\frac{16}{3} - \frac{10}{3} = \frac{6}{3}$$ yards. Choice B subtracts incorrectly. Choice C adds what she has to what she needs. Choice D is the total needed, not additional.

$$5 \times \frac{4}{7} = \frac{20}{7} = 20 \times \frac{1}{7}$$. This can also be written as $$5 \times 4 \times \frac{1}{7}$$ or $$4 \times 5 \times \frac{1}{7}$$ since multiplication is commutative. All three expressions equal $$\frac{20}{7}$$.

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 $$6 \times \frac{2}{9}$$.

If $$4 \times \frac{a}{5} = \frac{12}{5}$$, then $$\frac{4a}{5} = \frac{12}{5}$$, so $$4a = 12$$ and $$a = 3$$. This shows $$4 \times \frac{3}{5} = \frac{12}{5} = 12 \times \frac{1}{5}$$. Choice B uses wrong value. Choice C uses correct $$a$$ but wrong unit fraction form. Choice D uses the result as $$a$$.

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 2/3 is really 2 copies of the unit fraction 1/3, so 5 groups of (2 copies of 1/3) equals (5 × 2) copies of 1/3, which equals 10/3. To multiply 5 × (2/3), we multiply 5 × 2 = 10, and keep the denominator 3, giving 10/3. We can also see this as 5 groups of 2 unit fractions (1/3), which equals 10 unit fractions = 10 × (1/3) = 10/3. Choice A is correct because multiplying: 5 × 2 = 10, keeping denominator 3: 10/3. This demonstrates understanding that multiplying by whole number affects numerator only. Choice B represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 5. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 2/3 is '2 copies of 1/3,' so 5 groups of that is (5 × 2) copies of 1/3 = 10 × (1/3) = 10/3. Connect to repeated addition: 5 × (2/3) = 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 10/3. Convert improper fractions to mixed numbers for interpretation: 10/3 = 3 1/3.