When you need to compare fractions with different denominators, you can't just look at the numerators and denominators separately - you need to find a common denominator to make a fair comparison.

To compare $$\frac{3}{4}$$ and $$\frac{4}{5}$$, find the least common denominator. The denominators are 4 and 5, so you need a number that both divide into evenly. Since 4 × 5 = 20, use 20 as your common denominator.

Convert each fraction: $$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$$ and $$\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$$

Now you can easily compare: $$\frac{16}{20} > \frac{15}{20}$$, so $$\frac{4}{5} > \frac{3}{4}$$. Tommy ran farther on Tuesday.

Choice A makes a calculation error, incorrectly claiming that $$\frac{15}{20} > \frac{16}{20}$$ - this reverses the inequality. Choice B incorrectly states the fractions are equal when they clearly aren't after conversion. Choice D uses faulty reasoning by just adding numerators and denominators separately, which has nothing to do with comparing fraction values.

Remember this strategy: when comparing fractions with different denominators, always convert to a common denominator first. You can find the least common denominator by multiplying the denominators together if they don't share common factors, or find the least common multiple if they do.

To compare $$\frac{3}{8}$$ and $$\frac{2}{5}$$, find a common denominator. The LCD of 8 and 5 is 40. $$\frac{3}{8} = \frac{15}{40}$$ and $$\frac{2}{5} = \frac{16}{40}$$. Since $$\frac{16}{40} > \frac{15}{40}$$, Jake ate more pizza. Choice A incorrectly orders the fractions. Choice C is wrong because neither fraction equals $$\frac{1}{2}$$. Choice D incorrectly compares the number of slices without considering the different pizza divisions.

To compare $$\frac{2}{3}$$ and $$\frac{5}{9}$$, convert to a common denominator. Since 9 is a multiple of 3, use 9 as the common denominator. $$\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$$. Comparing $$\frac{6}{9}$$ and $$\frac{5}{9}$$: since $$6 > 5$$, we have $$\frac{6}{9} > \frac{5}{9}$$, so $$\frac{2}{3} > \frac{5}{9}$$. Choice B correctly converts but uses the wrong inequality symbol. Choice C incorrectly claims the fractions are equal. Choice D uses irrelevant arithmetic with the numbers.

First, compare each fraction to $$\frac{1}{2}$$. $$\frac{5}{12}$$ compared to $$\frac{6}{12} = \frac{1}{2}$$: since $$\frac{5}{12} < \frac{6}{12}$$, we have $$\frac{5}{12} < \frac{1}{2}$$. $$\frac{3}{8}$$ compared to $$\frac{4}{8} = \frac{1}{2}$$: since $$\frac{3}{8} < \frac{4}{8}$$, we have $$\frac{3}{8} < \frac{1}{2}$$. Both fractions are less than $$\frac{1}{2}$$. Using common denominator 24: $$\frac{5}{12} = \frac{10}{24}$$ and $$\frac{3}{8} = \frac{9}{24}$$, so $$\frac{5}{12} > \frac{3}{8}$$ and $$\frac{5}{12}$$ is closer to $$\frac{1}{2}$$. Choice A incorrectly states both are greater than $$\frac{1}{2}$$. Choices B and D incorrectly state $$\frac{3}{8} > \frac{1}{2}$$.

To compare $$\frac{5}{6}$$ and $$\frac{7}{9}$$, find a common denominator. The LCD of 6 and 9 is 18. $$\frac{5}{6} = \frac{15}{18}$$ and $$\frac{7}{9} = \frac{14}{18}$$. Since $$\frac{15}{18} > \frac{14}{18}$$, Emma needs more flour than she has. Choice A reverses the inequality. Choice C uses an incorrect common denominator and wrong conclusion. Choice D makes the error of comparing only numerators while ignoring denominators.

When comparing fractions, you need to find a common denominator to see which is actually larger. Since Maya and Carlos painted identical canvases, you're comparing $$\frac{5}{8}$$ and $$\frac{3}{4}$$ of the same size canvas.

To compare these fractions, convert them to the same denominator. Since 8 is a multiple of 4, use eighths as your common denominator. Maya already painted $$\frac{5}{8}$$, so no conversion needed. For Carlos: $$\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$$.

Now you can compare: $$\frac{5}{8}$$ versus $$\frac{6}{8}$$. Since $$\frac{6}{8} > \frac{5}{8}$$, Carlos painted more blue area. This makes choice D correct.

Choice A incorrectly states that $$\frac{5}{8} > \frac{3}{4}$$, but as we saw, $$\frac{5}{8} = \frac{5}{8}$$ while $$\frac{3}{4} = \frac{6}{8}$$, so this comparison is backwards. Choice B makes the mistake of only looking at the numerators (5 vs. 3) without considering that the denominators are different - you can't compare parts without knowing the size of the whole. Choice C incorrectly claims the fractions are equal, but $$\frac{5}{8}$$ and $$\frac{6}{8}$$ are clearly different amounts.

Remember: when comparing fractions, always convert to a common denominator first. Don't just compare numerators when denominators differ - the "pieces" aren't the same size!

To compare $$\frac{7}{10}$$ and $$\frac{2}{3}$$, find a common denominator. The LCD of 10 and 3 is 30. $$\frac{7}{10} = \frac{21}{30}$$ and $$\frac{2}{3} = \frac{20}{30}$$. Since $$\frac{21}{30} > \frac{20}{30}$$, Ben ate more of the first granola bar. Choice B correctly converts to equivalent fractions but reverses the inequality. Choice C incorrectly states the fractions are equal and mentions an impossible conversion. Choice D uses an irrelevant sum of numerators and denominators.

When comparing fractions with different denominators, you need to find a common denominator to see which fraction is actually larger. You can't just compare the numerators when the denominators are different.

To compare $$\frac{4}{7}$$ and $$\frac{3}{5}$$, find the least common denominator. Since 7 and 5 share no common factors, multiply them: 7 × 5 = 35. Now convert both fractions: $$\frac{4}{7} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35}$$ and $$\frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}$$. Since $$\frac{21}{35} > \frac{20}{35}$$, Lisa drank more water during science class.

Choice A makes the correct calculation but gets the comparison backwards—$$\frac{20}{35}$$ is not greater than $$\frac{21}{35}$$. Choice B incorrectly claims these fractions are equal; $$\frac{4}{7}$$ does not equal $$\frac{12}{21}$$ (which would be $$\frac{4}{7}$$), and $$\frac{3}{5}$$ definitely doesn't equal $$\frac{12}{21}$$ either. Choice C uses the dangerous shortcut of only comparing numerators—this only works when denominators are the same. Here, even though 4 > 3, we have $$\frac{4}{7} < \frac{3}{5}$$ because sevenths are smaller pieces than fifths.

Remember: when comparing fractions, always convert to a common denominator first. Never compare just the numerators unless the denominators are identical. This prevents the common trap of assuming that a larger numerator always means a larger fraction.

This question tests 4th grade ability to compare two fractions with different numerators and different denominators, using strategies like creating common denominators, common numerators, or comparing to benchmark fraction 1/2, recognizing comparisons are valid only when fractions refer to same whole (CCSS.4.NF.2). To compare fractions with different numerators and denominators, we can use several strategies. Common denominators: convert both fractions to the same denominator, then compare numerators (larger numerator = greater fraction). Common numerators: if numerators are the same, the fraction with the SMALLER denominator is GREATER (fewer parts means bigger pieces). Benchmark 1/2: compare each fraction to 1/2—if one is less than 1/2 and the other is greater than 1/2, you immediately know which is bigger. To compare 4/9 and 3/7, we can find common denominator 63, converting to 28/63 and 27/63, allowing direct comparison. Choice A is correct because using common denominators: 4/9 = 28/63 and 3/7 = 27/63, comparing numerators shows 28 > 27, so Jamal poured more. This demonstrates correct fraction comparison. Choice D represents assuming you cannot compare because denominators are different, which happens when students don't know conversion strategies. To help students: Practice all three strategies. For common denominators, find LCM or multiply denominators, convert both fractions, compare numerators. For common numerators (2/3 vs 2/5), emphasize: same numerator means same NUMBER of pieces, so smaller denominator = BIGGER pieces = greater fraction (thirds are bigger than fifths). For benchmark 1/2, teach how to identify: if numerator × 2 = denominator (or close), fraction is about 1/2. Use visual models with SAME-SIZED wholes to show why comparisons must use same whole. Number lines help visualize: farther right = greater. Watch for: reversing > and < symbols, comparing only numerators or only denominators without strategy, thinking larger denominator always means larger fraction, and not recognizing fractions must refer to same whole to compare.

This question tests 4th grade ability to compare two fractions with different numerators and different denominators, using strategies like creating common denominators, common numerators, or comparing to benchmark fraction $\frac{1}{2}$, recognizing comparisons are valid only when fractions refer to same whole (CCSS.4.NF.2). To compare fractions with different numerators and denominators, we can use several strategies. Common denominators: convert both fractions to the same denominator, then compare numerators (larger numerator = greater fraction). Common numerators: if numerators are the same, the fraction with the SMALLER denominator is GREATER (fewer parts means bigger pieces). Benchmark $\frac{1}{2}$: compare each fraction to $\frac{1}{2}$—if one is less than $\frac{1}{2}$ and the other is greater than $\frac{1}{2}$, you immediately know which is bigger. To compare $\frac{2}{9}$ and $\frac{1}{4}$, visual models show same-sized wholes with different shading, allowing direct comparison. Choice B is correct because visual shows more shaded in $\frac{1}{4}$, and using common denominators: $\frac{2}{9} = \frac{8}{36}$ and $\frac{1}{4} = \frac{9}{36}$, so $\frac{2}{9} < \frac{1}{4}$. This demonstrates correct fraction comparison. Choice A represents reversed symbol, which happens when students mix up symbol direction. To help students: Practice all three strategies. For common denominators, find LCM or multiply denominators, convert both fractions, compare numerators. For common numerators ($\frac{2}{3}$ vs $\frac{2}{5}$), emphasize: same numerator means same NUMBER of pieces, so smaller denominator = BIGGER pieces = greater fraction (thirds are bigger than fifths). For benchmark $\frac{1}{2}$, teach how to identify: if numerator × 2 = denominator (or close), fraction is about $\frac{1}{2}$. Use visual models with SAME-SIZED wholes to show why comparisons must use same whole. Number lines help visualize: farther right = greater. Watch for: reversing > and < symbols, comparing only numerators or only denominators without strategy, thinking larger denominator always means larger fraction, and not recognizing fractions must refer to same whole to compare.