To find the equivalent fraction, multiply both numerator and denominator by 3: $$\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$$. Choice B incorrectly adds 3 to the denominator instead of multiplying. Choice C represents a different amount entirely. Choice D only multiplies the denominator by 3.

To get from $$\frac{3}{8}$$ to $$\frac{12}{32}$$, both 3 and 8 must be multiplied by 4: $$3 \times 4 = 12$$ and $$8 \times 4 = 32$$. Choice B is the difference between 12 and 3. Choice C is the product of 3 and 8. Choice D is just the original denominator.

When you see equivalent fractions with different denominators, you're looking for fractions that represent the same amount but are written differently. Think of this like cutting the same pizza into different numbers of pieces - you still have the same amount of pizza.

To find equivalent fractions, you multiply or divide both the numerator and denominator by the same number. Here, you need to figure out what number transforms 9 into 27. Since $$27 ÷ 9 = 3$$, the denominator was multiplied by 3. This means you must also multiply the numerator by 3 to keep the fractions equivalent: $$\frac{5}{9} = \frac{5 × 3}{9 × 3} = \frac{15}{27}$$. The numerator of the second fraction is 15.

Let's check why the other answers don't work. Choice A (22) would give you $$\frac{22}{27}$$, which is much larger than $$\frac{5}{9}$$ - definitely not equivalent. Choice B (5) would give you $$\frac{5}{27}$$, which keeps the same numerator but makes the denominator bigger, creating a much smaller fraction than the original. Choice D (12) would give you $$\frac{12}{27}$$, which doesn't follow the rule of multiplying both parts by the same number.

Study tip: When finding equivalent fractions, always ask yourself "What did I multiply the denominator by?" Then multiply the numerator by that exact same number. This ensures both fractions represent the same amount, just divided into different-sized pieces.

When you see a question about equivalent fractions, you're looking for fractions that represent the same value but are written differently. To find equivalent fractions, you multiply or divide both the numerator and denominator by the same number.

Starting with $$\frac{4}{9}$$, you need to find what number to multiply 4 by to get 12. Since $$4 \times 3 = 12$$, you multiply the numerator by 3. To keep the fraction equivalent, you must multiply the denominator by the same number: $$9 \times 3 = 27$$. This gives you $$\frac{12}{27}$$, which equals the original $$\frac{4}{9}$$.

Let's check why the other answers don't work. Choice A (21) would give you $$\frac{12}{21}$$. If you simplify this by dividing both parts by 3, you get $$\frac{4}{7}$$, which is not equal to $$\frac{4}{9}$$. Choice C (36) creates $$\frac{12}{36}$$, which simplifies to $$\frac{1}{3}$$ when you divide by 12. This also doesn't equal $$\frac{4}{9}$$. Choice D (15) gives you $$\frac{12}{15}$$, which simplifies to $$\frac{4}{5}$$ when you divide by 3, again not matching our original fraction.

Remember this key strategy: whatever you multiply the numerator by to create an equivalent fraction, you must multiply the denominator by that exact same number. This keeps the fractions truly equivalent and prevents common mistakes.

When you're creating equivalent fractions, you multiply both the numerator (top number) and denominator (bottom number) by the same value. This keeps the fraction equal to the original while changing how it looks.

Let's follow Ben's work step by step. He starts with $$\frac{6}{8}$$ and multiplies both parts by 3. For the numerator: $$6 \times 3 = 18$$. For the denominator: $$8 \times 3 = 24$$. So Ben creates $$\frac{18}{24}$$, which is answer choice B.

You can verify this is equivalent to the original by checking that both fractions reduce to $$\frac{3}{4}$$ when simplified.

Now let's see why the other answers are wrong. Choice A ($$\frac{9}{11}$$) comes from a common mistake where someone multiplies the numerator by 3 correctly ($$6 \times 3 = 18$$... wait, that's 18, not 9) but then adds 3 to the denominator instead of multiplying ($$8 + 3 = 11$$). Actually, this answer involves multiple errors. Choice C ($$\frac{18}{11}$$) gets the numerator right but makes that addition error with the denominator. Choice D ($$\frac{9}{24}$$) correctly multiplies the denominator by 3 but incorrectly adds 3 to the numerator instead of multiplying.

The key strategy here is remembering that equivalent fractions require multiplying both the top AND bottom by the same number. If you multiply one part and add to the other, or use different numbers for each part, you'll get a completely different fraction that's not equivalent to your starting fraction.

This question tests 4th grade understanding of why a fraction a/b is equivalent to (n×a)/(n×b) by using visual fraction models, with attention to how number and size of parts differ even though the fractions represent the same amount (CCSS.4.NF.1). Equivalent fractions represent the same amount but have different numbers of parts. To generate an equivalent fraction, multiply both the numerator and denominator by the same whole number—this is like multiplying by n/n, which equals 1, so the value doesn't change. The key understanding: when you multiply both parts by the same number, you get MORE parts (denominator increases) but each part is SMALLER, so the total amount stays the same. The visual model on the number line shows 2/4 as halfway between 0 and 1 with 4 parts, and 4/8 also halfway with 8 parts—same amount, different partition, demonstrating equivalent fractions. Choice C is correct because both points are halfway between 0 and 1, showing the same value; this demonstrates understanding that multiplying top and bottom by the same number preserves the fraction's value. Choice D represents adding instead of multiplying, which happens when students think adding creates equivalence. To help students: Use visual models—show 2/4 and 4/8 on number lines where the points align at the same position but with different divisions. Emphasize the pattern: multiply both numerator AND denominator by the same number (from 1/2, multiply by 2 for 2/4, by 4 for 4/8).

When you're working with equivalent fractions, you're finding different ways to write the same amount by multiplying or dividing both the numerator and denominator by the same number. This keeps the fraction's value unchanged.

To find the equivalent fraction with denominator 50, you need to figure out what number to multiply 10 by to get 50. Since $$10 \times 5 = 50$$, you multiply both parts of $$\frac{7}{10}$$ by 5. This gives you $$\frac{7 \times 5}{10 \times 5} = \frac{35}{50}$$, which is answer choice C.

Let's see why the other answers don't work. Choice A gives $$\frac{14}{50}$$, which would mean you multiplied the numerator by 2 but the denominator by 5 – that's not allowed since you must use the same number for both parts. Choice B gives $$\frac{28}{50}$$, which would require multiplying by 4 in the numerator and 5 in the denominator – again, different numbers. Choice D gives $$\frac{42}{50}$$, which would mean multiplying by 6 in the numerator but 5 in the denominator.

You can verify that C is correct by checking if $$\frac{7}{10}$$ and $$\frac{35}{50}$$ represent the same amount: $$\frac{35}{50} = \frac{35 \div 5}{50 \div 5} = \frac{7}{10}$$ ✓

Remember: to create equivalent fractions, always multiply (or divide) the top and bottom by the exact same number. Find what number transforms your current denominator into the target denominator, then apply that same number to the numerator.

When you need to find an equivalent fraction with a different denominator, you're looking for a fraction that represents the same amount but is written differently. Think of it like having the same amount of pizza, but cutting it into more pieces.

To change $$\frac{4}{6}$$ to twelfths, you need to figure out what to multiply both the numerator and denominator by. Since you want the denominator to be 12, ask yourself: "What do I multiply 6 by to get 12?" The answer is 2, because $$6 \times 2 = 12$$.

Whatever you do to the denominator, you must do to the numerator to keep the fractions equivalent. So multiply the numerator by 2 as well: $$4 \times 2 = 8$$. This gives you $$\frac{8}{12}$$, which is answer choice D.

Let's see why the other answers are incorrect. Choice A ($$\frac{6}{12}$$) would mean someone just used 6 as the new numerator without doing the proper calculation. Choice B ($$\frac{4}{12}$$) happens when you change the denominator but forget to change the numerator at all. Choice C ($$\frac{16}{12}$$) occurs if you accidentally multiply 4 by 4 instead of 2, perhaps confusing this with a different conversion.

Here's your key strategy: always find the multiplication factor first by comparing denominators, then apply that same factor to the numerator. Remember the golden rule of equivalent fractions: whatever you do to the bottom, you must do to the top.

This question tests 4th grade understanding of why a fraction a/b is equivalent to (n×a)/(n×b) by using visual fraction models, with attention to how number and size of parts differ even though the fractions represent the same amount (CCSS.4.NF.1). Equivalent fractions represent the same amount but have different numbers of parts. To generate an equivalent fraction, multiply both the numerator and denominator by the same whole number—this is like multiplying by n/n, which equals 1, so the value doesn't change. The key understanding: when you multiply both parts by the same number, you get MORE parts (denominator increases) but each part is SMALLER, so the total amount stays the same. The visual model shows 1/2 has 1 part shaded out of 2 total, while equivalent fraction 3/6 has 3 parts shaded out of 6 total—same amount, different partition, demonstrating equivalent fractions. Choice C is correct because it explains the same amount is shaded, but the whole is cut into more, smaller pieces; this shows understanding that multiplying top and bottom by the same number preserves the fraction's value. Choice D represents adding instead of multiplying, which happens when students think adding creates equivalence. To help students: Use visual models—show 1/2 and 3/6 with area models where the SAME AMOUNT is shaded but with different numbers of parts. Emphasize the pattern: multiply both numerator AND denominator by the same number (if multiply by 3, do 1×3=3 AND 2×3=6, giving 3/6).

This question tests 4th grade understanding of why a fraction a/b is equivalent to (n×a)/(n×b) by using visual fraction models, with attention to how number and size of parts differ even though the fractions represent the same amount (CCSS.4.NF.1). Equivalent fractions represent the same amount but have different numbers of parts. To generate an equivalent fraction, multiply both the numerator and denominator by the same whole number—this is like multiplying by n/n, which equals 1, so the value doesn't change. The key understanding: when you multiply both parts by the same number, you get MORE parts (denominator increases) but each part is SMALLER, so the total amount stays the same. Starting with 4/9, multiplying numerator and denominator by 2 gives (4×2)/(9×2) = 8/18; the visual model shows 4/9 has 4 parts shaded out of 9 total, while equivalent fraction 8/18 has 8 parts shaded out of 18 total—same amount, different partition, demonstrating equivalent fractions. Choice A is correct because multiplying numerator by 2: 4×2=8, and denominator by 2: 9×2=18, giving 8/18; the visual models show the same amount shaded—4/9 and 8/18 cover the same portion of the whole. This shows understanding that multiplying top and bottom by the same number preserves the fraction's value. Choice B represents multiplying numerator only or an arithmetic error, which happens when students don't multiply both parts. To help students: Use visual models—show 4/9 and 8/18 with area models where the SAME AMOUNT is shaded but with different numbers of parts. Emphasize the pattern: multiply both numerator AND denominator by the same number (if multiply by 2, do 4×2=8 AND 9×2=18, giving 8/18); explain: more parts means each part is smaller, but total amount is the same; connect to multiplying by 1: multiplying by n/n (like 2/2 or 3/3) equals multiplying by 1, which doesn't change the value; show pattern: 4/9 = 8/18 = 12/27 = 16/36 (each time multiply by next whole number); watch for: multiplying only numerator or only denominator, adding instead of multiplying, using different numbers for top and bottom, and not understanding that MORE parts with SMALLER size equals SAME amount.