Create and Use Equivalent Ratio Tables - 6th Grade Math

1. A smoothie recipe uses a ratio of $3$ cups of yogurt to $4$ cups of fruit. Which table shows equivalent ratios for yogurt : fruit for $\times 1$ through $\times 5$?
   Answer: Yogurt: 3, 6, 9, 12, 15; Fruit: 4, 8, 12, 16, 20 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.
2. A snack mix uses a ratio of $3$ cups pretzels to $4$ cups cereal. Are the ratios $3:4$ and $12:16$ equivalent?
   Answer: Yes, because both terms are multiplied by $4$ 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 answer is choice B: Yes, because both terms are multiplied by 4, since 3×4=12 and 4×4=16. Common errors include checking differences like in A (additive thinking) or cross-products incorrectly like in C. To check equivalence, verify the scale factor is the same, like 12/3=4 and 16/4=4, or cross-multiply 3×16=48 and 4×12=48. Mistakes involve additive checks like adding instead of multiplying, or wrong arithmetic in cross-products.
3. Based on the coordinate plane shown, which equivalent ratio table correctly represents the plotted points?
   Answer: A table with x-values 3, 6, 9, 12 and corresponding y-values 2, 4, 6, 8 Reading the plotted points from the coordinate plane: (3,2), (6,4), (9,6), (12,8). This creates the ratio 3:2 and its equivalent ratios. The x-values are 3, 6, 9, 12 and y-values are 2, 4, 6, 8. Choice A reverses x and y values. Choice B uses a 1:2 ratio instead of 3:2. Choice D also reverses the coordinates.
4. Two snack mixes compare nuts to raisins. Mix A ratio (nuts:raisins) is $2:3$. Mix B ratio (nuts:raisins) is $3:5$. Using equivalent ratio tables or unit rates, which mix has the greater number of nuts per 1 raisin?
   Answer: Mix A This question tests comparing ratios using equivalent ratio tables or unit rates to determine which mix has more nuts per raisin. Equivalent ratios are formed by multiplying both parts by the same number, but here we compare unit rates: for Mix A 2:3, nuts per raisin is 2/3 ≈0.67, for Mix B 3:5 is 3/5=0.6. Tables can organize scaled versions, like for A: nuts 2,4,6 and raisins 3,6,9; for B: 3,6,9 and 5,10,15, to see rates. To compare, calculate unit rates (nuts/raisins) and see A's 0.67 > B's 0.6, so A has more nuts per raisin. The correct answer is Mix A, as it has the greater unit rate. Common errors include comparing totals without rates, like just looking at numerators, or inverting the ratios. For comparison, find unit rates by dividing; if scaling to same denominator (e.g., A to 10 raisins: nuts ≈6.67, B to 10: nuts=6), A is greater, with mistakes from not using a common basis or arithmetic errors.
5. A school is planning field trips using the ratio of 2 teachers to 15 students. The principal creates an equivalent ratio table but accidentally leaves some values blank. If one row shows 8 teachers, and another row shows 75 students, what is the sum of the missing values in these two rows?
   Answer: The sum of missing values is 70 because 60 students and 10 teachers are needed 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.
6. A map scale uses a ratio of $5$ cm to $2$ km. In an equivalent ratio table, the distance is $8$ km. What is the corresponding map length in cm?
   Answer: $20$ cm 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.
7. Three stores sell trail mix using different ratios of nuts to dried fruit. Store A uses 8:3, Store B uses 12:5, and Store C uses 16:6. A customer wants to buy the trail mix with the highest ratio of nuts to dried fruit. Which analysis correctly determines the best choice?
   Answer: Store C has the highest ratio at 2.67, followed by Store A at 2.67, then Store B at 2.40 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.
8. A store has a deal: $4$ notebooks cost $6$ dollars. Which ratio table shows an equivalent relationship between notebooks and cost (in dollars)?
   Answer: Notebooks: 4, 8, 12, 16; Cost: 6, 12, 18, 24 This question tests identifying equivalent ratio tables for notebooks to cost, scaled consistently from the original ratio. Equivalent ratios are created by multiplying both by the same factor; for 4:6 (simplifies to 2:3), scaling by ×2 gives 8:12, ×3 gives 12:18, ×4 gives 16:24, all with unit rate 2/3 notebooks per dollar. The table with notebooks:4,8,12,16 and cost:6,12,18,24 correctly shows this for ×1 to ×4. Choice B is correct, as each pair maintains 4:6 or simplifies to 2:3. Errors like additive increases (e.g., +4 to cost) appear in A and D, or mismatched quantities in C. Create the table by: (1) starting with 4:6, (2) scaling by factors, (3) listing in columns. Verify by checking if cost ÷ notebooks = 3/2 consistently across rows.
9. Are the ratios $6:8$ and $9:12$ equivalent?
   Answer: Yes, because both simplify to $3:4$. This question tests determining if ratios are equivalent by simplifying or cross-multiplying, which helps compare without tables. Equivalent ratios simplify to the same value; 6:8 simplifies to 3:4 (divide by 2), and 9:12 to 3:4 (divide by 3), or cross-products 6×12=72 and 8×9=72 are equal. A table for 3:4 would include 6:8, 9:12, confirming equivalence. Yes, they are equivalent because both simplify to 3:4. Errors include using addition (choice B) or misapplying cross-multiplication (choice C, though products are equal). To check: simplify both or compare cross-products. Avoid comparing sizes directly without simplifying, as in choice D.
10. A video game gives coins and gems in a ratio of $3:4$ (coins:gems). If a player earns $20$ gems, how many coins did they earn at the same rate?
    Answer: 15 This question tests using equivalent ratios to find a missing value, like coins for 20 gems in ratio 3:4, by scaling to match the given amount. Equivalent ratios are formed by multiplying both by the same number, like finding the factor for gems 4 to 20 (×5), then coins 3×5=15. A table can organize this: coins 3,6,9,12,15 and gems 4,8,12,16,20, showing the fifth row matches. For example, if 3:4 = ?:20, scale by ×5 to get 15:20. The correct number is 15 coins. Common errors include inverting the ratio or using addition, like adding 3 repeatedly. To solve, identify the scale factor from gems (4 to 20 is ×5), apply to coins; mistakes often involve wrong factors or not preserving the ratio.
11. A sports drink mix uses a ratio of $3$ scoops of powder for every $4$ cups of water. Which table shows equivalent ratios for this mixture and correctly fills in the missing value?
    Answer: Scoops: 3, 6, 9, 12, 15 Water: 4, 8, 12, 16, 20 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.
12. A science club mixes vinegar and baking soda in a ratio of $3:2$. They made a table of equivalent ratios and want to plot the ordered pairs $(\text{vinegar},\ \text{baking soda})$ on a coordinate plane. Which set of points is correct for 1–4 batches?
    Answer: $(3,2),\ (6,4),\ (9,6),\ (12,8)$ This question tests using equivalent ratio tables by scaling to generate ordered pairs for plotting on a coordinate plane, ensuring the points form a line through the origin for proportional relationships. Equivalent ratios are formed by multiplying both parts of the original ratio 3:2 (vinegar to baking soda) by the same number, like ×2 giving 6:4, ×3 giving 9:6, all preserving the ratio of 3/2. A table organizes these scaled versions, for example, vinegar 3,6,9,12 and baking soda 2,4,6,8, where each row is equivalent. For plotting, use pairs like (3,2), (6,4), (9,6), (12,8), which lie on a line through the origin since they are proportional. The correct set of points is option A, representing 1 to 4 batches accurately. Common errors include points that don't scale both by the same factor, like in option B with inconsistent increases, or option D with additive patterns. To plot, list the scaled pairs as coordinates and connect them; the line should pass through (0,0) for proportionality, with mistakes often from not verifying the scale factor or plotting non-equivalent ratios.
13. A school store sells pencils in packs. The ratio of pencils to dollars is $6:2$ (6 pencils cost $2). Which table correctly shows equivalent ratios?
    Answer: Pencils: 6,12,18,24; Dollars: 2,4,6,8 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.
14. A smoothie recipe uses a ratio of $3$ cups of yogurt to $4$ cups of fruit. Which table shows equivalent ratios for yogurt:fruit for 1–5 batches?
    Answer: Batches: 1,2,3,4,5; Yogurt (cups): 3,6,9,12,15; Fruit (cups): 4,8,12,16,20 This question tests creating equivalent ratio tables by scaling both quantities in the ratio by the same factor, such as for batches of a smoothie recipe, and ensures the tables maintain the proportional relationship without providing missing values or plotting here. Equivalent ratios are formed by multiplying both parts of the original ratio 3:4 (yogurt to fruit) by the same number, like ×2 giving 6:8, ×3 giving 9:12, all preserving the ratio of 3/4. A table organizes these scaled versions, for example, for ratio 3:4 across 1 to 5 batches: yogurt 3,6,9,12,15 and fruit 4,8,12,16,20, where each row is equivalent. For instance, if you have 3:4 and scale to find equivalents, ×4 gives 12:16, fitting the pattern in the table. The correct table is option A, which properly scales both quantities by 1 through 5 for each batch. Common errors include not scaling both parts equally, like in option B where yogurt adds 2 instead of multiplying, or option C where fruit adds 2, breaking the ratio. To create such a table, start with the given ratio 3:4, scale by factors 1 to 5 to get the pairs, and organize in columns for batches, yogurt, and fruit; mistakes often involve additive thinking instead of multiplicative scaling.
15. Two different juice mixtures are being compared. Mixture A uses 6 cups of apple juice to 4 cups of cranberry juice. Mixture B uses 9 cups of apple juice to 5 cups of cranberry juice. To determine which mixture has a stronger apple flavor, Sarah needs to compare the ratios. What can she conclude?
    Answer: Mixture B has a stronger apple flavor because 9:5 equals 1.8:1, while 6:4 equals 1.5:1 in apple-to-cranberry 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.
16. Two smoothie recipes are compared. Recipe A: $5$ strawberries for every $2$ cups of yogurt Recipe B: $6$ strawberries for every $3$ cups of yogurt Using equivalent ratios or unit rates, which recipe uses more strawberries per 1 cup of yogurt?
    Answer: Recipe A 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 answer is choice A: Recipe A uses more strawberries per yogurt, with unit rate 5/2=2.5 vs B's 6/3=2. Common errors include thinking they are equal since both simplify to similar but actually different rates, or comparing without units. To compare, calculate unit rates: strawberries per yogurt 5/2 > 6/3. Mistakes involve comparison without common basis, like totaling instead of per unit, or suggesting a graph is needed when rates suffice.
17. Using the table shown, which statement correctly compares the ratios of red marbles to blue marbles across different containers?
    Answer: Container C has the highest ratio, and Container A has the lowest ratio of red to blue marbles Calculate each ratio: Container A: 6:9 = 2:3 ≈ 0.67, Container B: 8:10 = 4:5 = 0.8, Container C: 12:8 = 3:2 = 1.5. Container C has the highest ratio (1.5) and Container A has the lowest (0.67). Choice A reverses the comparison. Choice B incorrectly identifies Container B as highest. Choice D incorrectly identifies Container B as lowest.
18. A recipe uses $4$ cups of flour for every $7$ cups of water. Are the ratios $4:7$ and $12:21$ equivalent?
    Answer: Yes, because both numbers were multiplied by $3$ This question tests understanding equivalent ratios without a full table, by checking if scaling the original 4:7 (flour to water) by a factor like 3 gives 12:21. Equivalent ratios are formed by multiplying both by the same number, preserving the relationship, like 4×3=12 and 7×3=21, both equal to 4/7. While tables can organize multiples, here it's direct: the pairs are equivalent since they scale by 3. For example, a table would show 4:7, 8:14, 12:21, all proportional. The correct answer is yes, because both were multiplied by 3, as in option B. Common errors include checking addition like in A or subtraction in C, instead of multiplication. To verify, find the scale factor (4 to 12 is ×3), apply to the other (7×3=21); mistakes arise from not recognizing multiplicative scaling or confusing with simplification.
19. A school club makes bracelets using $4$ blue beads for every $3$ red beads. Which ordered pairs $(\text{red},\text{blue})$ should be plotted to represent the equivalent ratios for 1–4 groups of the basic ratio?
    Answer: $(3,4),\ (6,8),\ (9,12),\ (12,16)$ 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 ordered pairs are in choice A: (3,4), (6,8), (9,12), (12,16), representing red:blue as 3:4 scaled by ×1 to ×4. Common errors include switching the order like in B to (4,3) etc., or not scaling properly like in C with additive increases. For plotting, use the pairs as coordinates, ensuring they form a line through the origin in proportional y=kx form. Mistakes involve plot errors like not passing through origin or incorrect scaling, such as in D with non-equivalent ratios.
20. Two buses travel at constant rates. Bus A travels $9$ miles in $3$ hours. Bus B travels $12$ miles in $5$ hours. Using ratio tables (miles per hour), which bus has the greater speed?
    Answer: Bus A This question tests comparing ratios using tables or unit rates to find which bus has greater speed in miles per hour. Equivalent ratios are formed by multiplying, but here we find unit rates: Bus A 9:3 =3 mph, Bus B 12:5=2.4 mph. Tables can scale to compare, like A: miles 9,18,27 and hours 3,6,9 (3 mph); B: 12,24,36 and 5,10,15 (2.4 mph). To compare, calculate rates (miles/hours), showing A's 3 > B's 2.4. The bus with greater speed is Bus A. Common errors include comparing totals without dividing, like thinking more miles means faster without time. For comparison, compute unit rates; scaling to same hours (e.g., A in 5 hours: 15 miles, B 12 miles) shows A faster, with mistakes from not using rates or arithmetic errors.

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