First, find the sugar needed for each batch using the ratio 3:2 (flour:sugar). For 18 cups flour: 18÷3=6, so 6×2=12 cups sugar. For 24 cups flour: 24÷3=8, so 8×2=16 cups sugar. For 30 cups flour: 30÷3=10, so 10×2=20 cups sugar. Total sugar = 12+16+20 = 48 cups. Choice A incorrectly uses 2:1 ratio. Choice C adds the flour amounts instead of calculating sugar properly. Choice D incorrectly uses 1:1 ratio.

When comparing ratios to determine which mixture has a stronger flavor, you need to find out how much of one ingredient there is for every unit of the other. The key is converting each ratio to see which has more apple juice per cup of cranberry juice.

For Mixture A (6 cups apple to 4 cups cranberry), divide both numbers by 4: $$6 ÷ 4 = 1.5$$ and $$4 ÷ 4 = 1$$. This gives you 1.5:1, meaning 1.5 cups of apple juice for every 1 cup of cranberry juice.

For Mixture B (9 cups apple to 5 cups cranberry), divide both by 5: $$9 ÷ 5 = 1.8$$ and $$5 ÷ 5 = 1$$. This gives you 1.8:1, meaning 1.8 cups of apple juice for every 1 cup of cranberry juice.

Since 1.8 > 1.5, Mixture B has more apple juice per cup of cranberry juice, making it stronger in apple flavor. Answer D correctly identifies this reasoning.

Answer A incorrectly focuses on whether ratios can be simplified rather than comparing their actual values. Answer B makes the mistake of thinking fewer total cups means stronger concentration—but concentration depends on the ratio, not the total amount. Answer C incorrectly assumes that any mixture with more apple than cranberry has equal apple strength, ignoring the specific proportions.

Remember: when comparing ratios, convert them to the same format (like "something to 1") so you can easily see which is larger. This makes ratio comparisons much clearer.

Using ratio 2:15 (teachers:students). For 8 teachers: 8÷2=4, so 4×15=60 students needed. For 75 students: 75÷15=5, so 5×2=10 teachers needed. Missing values: 60 students + 10 teachers = 70 total. Choice B miscalculates the teachers needed for 75 students. Choice C uses wrong student calculation. Choice D uses incorrect calculations for both scenarios.

Using ratio 5:3 (vegetables:meat), find meat portions: For 25 vegetables: 25÷5=5, so 5×3=15 meat (x=15). For 40 vegetables: 40÷5=8, so 8×3=24 meat (y=24). For 45 vegetables: 45÷5=9, so 9×3=27 meat (z=27). Therefore x+y+z = 15+24+27 = 66. Choice A uses incorrect calculations. Choice C adds an extra 3 to the total. Choice D miscalculates one or more ratios.

When comparing ratios, you need to convert each ratio to a decimal or fraction to see which is actually largest. The ratio "nuts to dried fruit" means nuts ÷ dried fruit.

Let's calculate each store's ratio:

• Store A: $$8 ÷ 3 = 2.67$$
• Store B: $$12 ÷ 5 = 2.40$$
• Store C: $$16 ÷ 6 = 2.67$$

Store C has the highest ratio at 2.67, meaning it has 2.67 parts nuts for every 1 part dried fruit. Store A ties for second place with the same ratio of 2.67, while Store B has the lowest ratio at 2.40.

Choice A incorrectly lists Store A as having the highest ratio when it actually ties with Store C. More importantly, it shows Store C at 2.67 in second place when Store C should be first. Choice B makes similar ranking errors and incorrectly places Store B in the middle position. Choice D completely reverses the order, putting Store B (the lowest ratio) first and showing the two tied stores as if they have different rankings.

The key trap here is thinking that larger numbers in the original ratio automatically mean a higher ratio. Store B uses 12:5, which might look "bigger" than 8:3, but when you do the division, 12÷5 = 2.40 is actually smaller than 8÷3 = 2.67.

Always convert ratios to decimals when comparing them. Don't be fooled by which numbers look larger in the original ratio format.

This question tests creating equivalent ratio tables by scaling both quantities in the ratio 6:2 (pencils to dollars) by the same factor for multiple packs. Equivalent ratios are formed by multiplying both by the same number, like ×2 giving 12:4, ×3 giving 18:6, all preserving the simplified ratio of 3:1. A table organizes these scaled versions, for example, pencils 6,12,18,24 and dollars 2,4,6,8, where each row is equivalent by ×1 to ×4. For instance, scaling 6:2 by ×4 gives 24:8, fitting the pattern. The correct table is option B, which properly scales both by 1 through 4. Common errors include inconsistent scaling, like option A with dollars not multiplying correctly, or option C with pencils adding 4. To create the table, start with 6:2, scale by factors 1 to 4, and list in columns; mistakes often involve simplifying incorrectly or using addition instead of multiplication.

This question tests creating equivalent ratio tables by scaling both quantities by the same factor, finding missing values, plotting on coordinate planes through the origin, and comparing ratios. Equivalent ratios are formed by multiplying both parts of the ratio by the same number to preserve the relationship, such as 3:4 scaled by ×2 gives 6:8, by ×3 gives 9:12, all equal to the ratio 3/4; a table organizes these scaled versions, like for ratio 2:3 with batches 1,2,3,4 and cups 2,4,6,8 where each row is equivalent to 2:3; finding missing values involves identifying the scale, like if 3:4=9:?, the scale from 3 to 9 is ×3, so 4×3=12; plotting pairs like (1,2), (2,4), (3,6) on a plane forms a line through the origin indicating proportionality; comparing ratios uses unit rates, like 2:3 is 2/3≈0.67 and 3:5 is 0.6, so the first is greater. For example, with ratio 3:4, a table could be scoops 3,6,9,12 and water 4,8,12,16 (scaled by ×1,×2,×3,×4), and for missing if 3:4=?:12, the scale is ×3 so ?=9; plotting (3,4), (6,8), (9,12) forms a line through (0,0). The correct table is choice B: scoops 3,6,9,12,15 and water 4,8,12,16,20, as it consistently scales 3:4 by ×1 to ×5. Common errors include not scaling both quantities equally, like in A where water is 4,8,10,16,20 using additive thinking instead of multiplication, or in C with scoops 3,5,7,9,11 not multiplying by the same factor. To create such a table, start with the ratio 3:4, scale by factors like ×1,×2,×3,×4,×5 to get 3:4, 6:8, 9:12, 12:16, 15:20, and organize in columns. Mistakes often involve additive scaling like adding 3 and 4 repeatedly instead of multiplying, leading to incorrect tables like in choices A, C, or D.

This question tests creating equivalent ratio tables by scaling both quantities by the same factor, finding missing values, plotting on coordinate planes through the origin, and comparing ratios. Equivalent ratios are formed by multiplying both parts of the ratio by the same number to preserve the relationship, such as 3:4 scaled by ×2 gives 6:8, by ×3 gives 9:12, all equal to the ratio 3/4; a table organizes these scaled versions, like for ratio 2:3 with batches 1,2,3,4 and cups 2,4,6,8 where each row is equivalent to 2:3; finding missing values involves identifying the scale, like if 3:4=9:?, the scale from 3 to 9 is ×3, so 4×3=12; plotting pairs like (1,2), (2,4), (3,6) on a plane forms a line through the origin indicating proportionality; comparing ratios uses unit rates, like 2:3 is 2/3≈0.67 and 3:5 is 0.6, so the first is greater. For example, with ratio 3:4, a table could be 3,6,9,12 | 4,8,12,16 (×1,×2,×3,×4), missing: if 3:4=?:12, find 3×3=9; plot (3,4), (6,8), (9,12) line through (0,0). The correct missing value is ?=8 in choice B, since the table shows flour 2,4,6,? and milk 5,10,15,20, scaled by ×1,×2,×3,×4, so 2×4=8 matches 5×4=20. A common error is using additive thinking, like adding 2 repeatedly to get ?=10 in C, instead of multiplying by the scale factor of 4 from 5 to 20. To find missing values, identify the scale factor, such as from 5 to 20 is ×4, then apply to flour: 2×4=8. Mistakes include wrong scale factor, like dividing instead of multiplying, or arithmetic errors leading to ?=7,10, or 12.

This question tests creating equivalent ratio tables by scaling both quantities by the same factor, finding missing values, plotting on coordinate planes through the origin, and comparing ratios. Equivalent ratios are formed by multiplying both parts of the ratio by the same number to preserve the relationship, such as 3:4 scaled by ×2 gives 6:8, by ×3 gives 9:12, all equal to the ratio 3/4; a table organizes these scaled versions, like for ratio 2:3 with batches 1,2,3,4 and cups 2,4,6,8 where each row is equivalent to 2:3; finding missing values involves identifying the scale, like if 3:4=9:?, the scale from 3 to 9 is ×3, so 4×3=12; plotting pairs like (1,2), (2,4), (3,6) on a plane forms a line through the origin indicating proportionality; comparing ratios uses unit rates, like 2:3 is 2/3≈0.67 and 3:5 is 0.6, so the first is greater. For example, with ratio 3:4, a table could be 3,6,9,12 | 4,8,12,16 (×1,×2,×3,×4), missing: if 3:4=?:12, find 3×3=9; plot (3,4), (6,8), (9,12) line through (0,0). The correct map length is 20 cm in choice C, since scale 5 cm:2 km, for 8 km the factor is 8/2=4, so 5×4=20 cm. Common errors include wrong scale factor, like multiplying by 8/5 instead, or arithmetic mistakes leading to 16 cm or 40 cm. To find missing values, find scale factor like from 2 km to 8 km is ×4, apply to 5 cm×4=20. Mistakes include additive scaling instead of multiplying, or confusing which quantity to scale.

This question tests creating equivalent ratio tables by scaling both quantities in the ratio by the same factor, such as multiplying by 1 through 5, and identifying the correct table that maintains the proportional relationship. Equivalent ratios are formed by multiplying both parts of the original ratio by the same number, preserving the relationship; for example, the ratio 3:4 scaled by ×2 gives 6:8, which is equivalent, and scaling by ×3 gives 9:12, all equal to the unit rate of 3/4. A table organizes these scaled versions; for the yogurt to fruit ratio of 3:4, the table should show yogurt amounts like 3, 6, 9, 12, 15 and corresponding fruit amounts of 4, 8, 12, 16, 20 for ×1 through ×5, with each row maintaining the 3:4 ratio. In this case, choice B correctly shows yogurt: 3,6,9,12,15 and fruit: 4,8,12,16,20, as each pair is scaled consistently from the original ratio. Common errors include additive scaling instead of multiplicative, like in choice A where fruit increases by +3 each time, or mismatched scaling as in choice C and D, leading to non-equivalent ratios. To create such a table: (1) start with the given ratio (3:4), (2) scale by factors ×1 to ×5 to get pairs like (3,4), (6,8), etc., (3) organize in rows or columns. Always verify equivalence by checking if the ratios simplify to the same value or by cross-multiplying to confirm consistency across rows.