Multi-Operation Problems - SSAT Middle Level Quantitative

1. A car rental company charges $25 per day plus $0.15 per mile driven. If someone rents a car for 3 days and drives 240 miles, what is the total cost?
   Answer: $111.00 This is a multi-step word problem involving linear cost functions, where you need to calculate both fixed costs (daily charges) and variable costs (mileage charges), then add them together. Start by identifying the two cost components: the daily rental fee of $25 per day, and the mileage charge of $0.15 per mile. For a 3-day rental, the daily charges are $3 \times \$25 = \$75$. For 240 miles driven, the mileage charges are $240 \times \$0.15 = \$36$. The total cost is $\$75 + \$36 = \$111$. Choice A ($75.00) represents only the daily rental charges without including any mileage fees—a common error when students forget about the variable cost component. Choice C ($114.00) likely comes from miscalculating the mileage portion, perhaps computing $240 \times \$0.20 = \$48$ instead of the correct $240 \times \$0.15 = \$36$, then adding $\$75 + \$48 = \$123$... actually, this doesn't match C either, but it's close to the right answer and might result from arithmetic errors. Choice D ($116.00) could result from adding an extra day or making calculation mistakes in the mileage computation. When tackling rental or pricing problems, always break them into fixed costs (time-based charges) and variable costs (usage-based charges). Double-check your arithmetic, especially with decimal multiplication, and make sure you've included all cost components before selecting your answer.
2. A factory produces 150 widgets per hour for the first 6 hours, then increases production to 200 widgets per hour for the next 4 hours. What is the average number of widgets produced per hour over the entire 10-hour period?
   Answer: 170 widgets per hour When you encounter questions about average rates over different time periods, you need to find the total output divided by total time, not simply average the two rates. First, calculate the total widgets produced. In the first 6 hours at 150 widgets per hour: $6 \times 150 = 900$ widgets. In the next 4 hours at 200 widgets per hour: $4 \times 200 = 800$ widgets. Total production is $900 + 800 = 1,700$ widgets over 10 hours. The average production rate is: $\frac{1,700 \text{ widgets}}{10 \text{ hours}} = 170$ widgets per hour. Answer A (170 widgets per hour) is correct because it properly accounts for the weighted average based on time periods. Answer B (175 widgets per hour) likely comes from simply averaging the two rates: $\frac{150 + 200}{2} = 175$. This ignores that the factory operated at each rate for different amounts of time. Answer C (180 widgets per hour) might result from incorrectly weighting the rates or making calculation errors in the total production. Answer D (185 widgets per hour) is too high and suggests confusion about how to handle the different time periods or rates. Remember: when calculating averages across different time periods with different rates, you must use a weighted average. Calculate total output, then divide by total time. Simply averaging the rates themselves will mislead you unless the time periods are equal.
3. A train travels 180 miles in 3 hours. At this same rate, how long will it take to travel 420 miles?
   Answer: 7 hours When you encounter a problem about constant rates, you're dealing with proportional relationships. The key insight is that if something travels at a steady rate, the ratio of distance to time remains the same. First, find the train's speed by dividing the distance by time: $\frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ miles per hour}$. Now you can find how long it takes to travel 420 miles at this same rate: $\frac{420 \text{ miles}}{60 \text{ miles per hour}} = 7 \text{ hours}$. Alternatively, you can set up a proportion: $\frac{180 \text{ miles}}{3 \text{ hours}} = \frac{420 \text{ miles}}{x \text{ hours}}$. Cross-multiplying gives you $180x = 1260$, so $x = 7$ hours. Choice A (6 hours) would mean the train is traveling 70 mph, which is faster than the given rate. Choice C (8 hours) suggests a speed of 52.5 mph, slower than the actual rate. Choice D (9 hours) implies an even slower speed of about 46.7 mph. These answers likely come from calculation errors or misunderstanding the relationship between the given information and what you're solving for. Choice B (7 hours) correctly maintains the 60 mph rate established in the problem. Remember that rate problems often test your ability to recognize when quantities are directly proportional. Always check that your answer makes sense—if the distance increases, time should increase proportionally when speed stays constant.
4. Sarah buys 4 pounds of apples at $2.50 per pound and 3 pounds of oranges at $3.20 per pound. If she pays with a $20 bill, how much change will she receive?
   Answer: $0.40 When you encounter multi-step word problems involving money, break them down systematically: calculate the total cost, then find the change from the amount paid. First, calculate the cost of each item separately. Sarah's apples cost $4 \text{ pounds} \times \$2.50 = \$10.00$. Her oranges cost $3 \text{ pounds} \times \$3.20 = \$9.60$. The total purchase is $\$10.00 + \$9.60 = \$19.60$. Since Sarah pays with a $20 bill, her change is $\$20.00 - \$19.60 = \$0.40$, making A correct. Let's examine why the other answers are wrong. Choice B ($0.60) likely comes from miscalculating one of the products—perhaps computing the oranges as $9.40 instead of $9.60, giving a total of $19.40 and change of $0.60. Choice C ($1.40) suggests a significant calculation error, possibly from getting a total cost of $18.60, which could happen if you mistakenly used $2.20 instead of $3.20 for the orange price. Choice D ($1.60) might result from calculating the apple cost as $9.00 (using $2.25 instead of $2.50) and getting a total of $18.40. The key strategy here is to work methodically and double-check your arithmetic, especially with decimal multiplication. Write out each step clearly: item cost calculations, total cost, then subtraction for change. Money problems on the SSAT often include answer choices that result from common calculation mistakes, so careful computation is essential.
5. A store sells notebooks in packs of 5 and pens in packs of 8. What is the smallest number of packs of each that must be bought to have an equal number of notebooks and pens?
   Answer: 8 packs of notebooks and 5 packs of pens This question tests your understanding of the least common multiple (LCM) - finding the smallest number that two different numbers can both divide into evenly. When you see problems asking for equal quantities from different pack sizes, you need to find the LCM of those pack sizes. Notebooks come in packs of 5, and pens come in packs of 8. To have equal numbers of each item, you need to find the smallest number that's divisible by both 5 and 8. Since 5 and 8 share no common factors (they're relatively prime), their LCM is simply $5 \times 8 = 40$. So you need 40 notebooks and 40 pens. This means $40 ÷ 5 = 8$ packs of notebooks and $40 ÷ 8 = 5$ packs of pens. Looking at the wrong answers: Choice A gives you $5 \times 5 = 25$ notebooks and $8 \times 8 = 64$ pens - these aren't equal. Choice B gives you $6 \times 5 = 30$ notebooks and $5 \times 8 = 40$ pens - again, not equal. Choice D gives you $10 \times 5 = 50$ notebooks and $8 \times 8 = 64$ pens - still unequal. These wrong answers might tempt you if you simply swap the pack numbers or use the original pack sizes as your answer. Choice C correctly gives you $8 \times 5 = 40$ notebooks and $5 \times 8 = 40$ pens. Remember: when finding equal quantities from different pack sizes, always calculate the LCM first, then divide by each pack size to find how many packs you need.
6. A swimming pool is being filled with water. After 2 hours, the pool is 1/4 full. After 5 hours total, the pool is 5/8 full. At this rate, how many more hours will it take to completely fill the pool?
   Answer: 3 hours When you encounter rate problems involving filling or emptying, focus on finding the consistent rate of change per unit time. This type of problem tests your ability to work with rates and proportional reasoning. Let's determine the pool's filling rate by examining what happened between hours 2 and 5. In those 3 hours, the pool went from $\frac{1}{4}$ full to $\frac{5}{8}$ full. The change was $\frac{5}{8} - \frac{1}{4} = \frac{5}{8} - \frac{2}{8} = \frac{3}{8}$ of the pool. Since this $\frac{3}{8}$ was filled in 3 hours, the rate is $\frac{3/8}{3} = \frac{1}{8}$ of the pool per hour. After 5 hours, $\frac{5}{8}$ of the pool is full, meaning $1 - \frac{5}{8} = \frac{3}{8}$ remains to be filled. At a rate of $\frac{1}{8}$ pool per hour, it will take $\frac{3/8}{1/8} = 3$ more hours to finish filling. Choice A (2 hours) would only fill $\frac{2}{8} = \frac{1}{4}$ more of the pool, leaving it $\frac{7}{8}$ full. Choice C (4 hours) assumes a slower rate and would overfill the pool. Choice D (5 hours) significantly overestimates the time needed, perhaps confusing total time with additional time. Study tip: In rate problems, always identify what changed over a specific time interval to find the consistent rate, then use that rate to solve for the unknown. Don't assume the rate from just the starting condition—look for the pattern between two data points.
7. A store offers a 20% discount on all items, then applies a 6% sales tax to the discounted price. If an item originally costs $45, what is the final price after the discount and tax?
   Answer: $38.16 When you encounter problems involving sequential percentage changes like discounts followed by taxes, you must apply each percentage to the result of the previous calculation—never to the original amount for both steps. Let's work through this step-by-step. First, apply the 20% discount to the original $45 price. A 20% discount means you pay 80% of the original price: $45 \times 0.80 = 36$. So the discounted price is $36. Next, apply the 6% sales tax to this discounted price of $36. Sales tax is added to the price, so you'll pay 106% of the discounted amount: $36 \times 1.06 = 38.16$. The final price is $38.16. Looking at the wrong answers: Choice A ($33.48) represents applying both the discount AND treating the tax as a reduction rather than an addition—a double error. Choice B ($36.00) is the price after discount but before tax is applied. Choice D ($41.40) comes from incorrectly applying the 6% tax to the original $45 price, then subtracting the 20% discount from that result. The key strategy for percentage problems is to work sequentially: complete each step fully before moving to the next, and always apply the second percentage to the result of the first calculation, not the original number. Watch for answer choices that represent intermediate steps—they're common traps on the SSAT.
8. A school fundraiser sells tickets for $8 each. The cost to rent the venue is $200, and refreshments cost $3 per person attending. If 150 people attend, what is the profit from the fundraiser?
   Answer: $550 When you encounter profit problems, remember that profit equals total revenue minus total costs. You need to identify all sources of income and all expenses to find the net result. Let's calculate the profit step by step. First, find the total revenue: 150 people × $8 per ticket = $1,200. Next, identify all costs: the venue rental is a fixed cost of $200, and refreshments cost $3 per person, so 150 people × $3 = $450 in refreshment costs. Total costs = $200 + $450 = $650. Therefore, profit = $1,200 (revenue) - $650 (total costs) = $550. Looking at the wrong answers: Choice A ($250) likely comes from forgetting the venue cost entirely—just calculating $1,200 revenue minus $450 refreshments minus an arbitrary $500. Choice B ($350) might result from a calculation error, perhaps subtracting the venue cost twice or making an arithmetic mistake when combining costs. Choice C ($450) represents a common trap where students calculate only the refreshment costs instead of the actual profit, confusing an expense category with the final answer. The key strategy for profit problems is to organize your work clearly: write "Revenue:" and list all income sources, then "Costs:" and list all expenses. Many students rush and miss either fixed costs (like venue rental) or variable costs (like per-person refreshments). Double-check that you're subtracting total costs from total revenue, not just one type of cost.
9. A recipe calls for 2.5 cups of flour to make 18 cookies. How many cups of flour are needed to make 45 cookies?
   Answer: 6.25 cups This is a proportion problem where you need to find how much of one ingredient is needed when scaling a recipe up or down. When you see questions about recipes, rates, or "per unit" relationships, think proportions. Set up the proportion by comparing cups of flour to number of cookies. You know that 2.5 cups makes 18 cookies, and you need to find how many cups make 45 cookies. Write this as: $\frac{2.5 \text{ cups}}{18 \text{ cookies}} = \frac{x \text{ cups}}{45 \text{ cookies}}$ Cross multiply to solve: $2.5 \times 45 = 18 \times x$, which gives you $112.5 = 18x$. Dividing both sides by 18: $x = \frac{112.5}{18} = 6.25$ cups. Looking at the wrong answers: Choice A (5.25 cups) is too small—this might come from incorrectly setting up the proportion or making an arithmetic error in the division. Choice C (6.75 cups) could result from adding instead of using proper proportion methods, or from calculation mistakes. Choice D (7.25 cups) is significantly too high and likely comes from confusing the setup of the ratio or major computational errors. You can double-check by seeing if the ratio makes sense: you're making 2.5 times as many cookies (45 ÷ 18 = 2.5), so you should need 2.5 times as much flour (2.5 × 2.5 = 6.25). Strategy tip: For proportion problems, always check if your answer makes logical sense. If you're making more of something, you should need more ingredients—and the scaling factor should be consistent.
10. In a school election, candidate A received 2/5 of the votes, candidate B received 1/3 of the votes, and candidate C received the remaining votes. If 450 students voted, how many votes did candidate C receive?
    Answer: 120 votes When you encounter fraction problems involving "remaining" amounts, you're working with parts of a whole that must add up to 1 (or 100%). The key is finding what's left after accounting for the known portions. First, find what fraction of votes candidates A and B received together. Candidate A got $\frac{2}{5}$ and candidate B got $\frac{1}{3}$. To add these fractions, find a common denominator: $\frac{2}{5} = \frac{6}{15}$ and $\frac{1}{3} = \frac{5}{15}$. Together they received $\frac{6}{15} + \frac{5}{15} = \frac{11}{15}$ of the votes. Since all votes must total 1, candidate C received $1 - \frac{11}{15} = \frac{4}{15}$ of the votes. With 450 total votes, candidate C got $\frac{4}{15} \times 450 = \frac{1800}{15} = 120$ votes. Looking at the wrong answers: Choice A (90 votes) represents $\frac{1}{5}$ of the total votes, which might result from incorrectly calculating the remaining fraction. Choice C (150 votes) equals $\frac{1}{3}$ of the votes—you might get this by confusing candidate C's share with candidate B's share. Choice D (180 votes) represents $\frac{2}{5}$ of the votes, which would happen if you confused candidate C with candidate A. The correct answer is B: 120 votes. Strategy tip: In "remaining amount" problems, always verify your fractions add up to 1 before calculating. This catches arithmetic errors early and ensures you're working with the right proportions.
11. A bakery makes muffins and sells them for $2.50 each. The daily fixed costs are $45, and each muffin costs $0.80 to make. How many muffins must be sold in one day to make a profit of at least $85?
    Answer: 77 muffins When you encounter profit problems, you need to set up an equation using the relationship: Profit = Revenue - Total Costs. Here, you're looking for the number of muffins needed to achieve at least $85 in profit. Let's define the variable: let $n$ = number of muffins sold. Revenue from selling $n$ muffins = $2.50n$ Total costs = Fixed costs + Variable costs = $45 + 0.80n$ Setting up the profit equation: $\text{Profit} = 2.50n - (45 + 0.80n) = 2.50n - 45 - 0.80n = 1.70n - 45$ For a profit of at least $85: $1.70n - 45 \geq 85$ $1.70n \geq 130$ $n \geq 76.47$ Since you can't sell a fraction of a muffin, you need to round up to 77 muffins. Let's verify: With 77 muffins, profit = $1.70(77) - 45 = 130.90 - 45 = 85.90$, which exceeds $85. Choice A (72 muffins) gives a profit of $1.70(72) - 45 = 77.40$, which falls short of $85. Choice B (76 muffins) yields $1.70(76) - 45 = 84.20$, still below the target. Choice D (80 muffins) would work but represents unnecessary overproduction since the question asks for the minimum needed. The answer is C: 77 muffins. Remember: In "at least" problems, always round up to the next whole number when dealing with discrete items, and verify your answer by substituting back into the original inequality.
12. Jenny runs at a constant speed of 8 mph for 45 minutes, then walks at 3 mph for 30 minutes. What is her average speed for the entire trip?
    Answer: 6.0 mph When you encounter average speed problems, remember that average speed isn't simply the arithmetic mean of the speeds—you need to consider both distance and time for the entire journey. First, convert the times to hours: 45 minutes = 0.75 hours and 30 minutes = 0.5 hours. Next, calculate the distance for each segment using $\text{distance} = \text{speed} \times \text{time}$: - Running: $8 \text{ mph} \times 0.75 \text{ hours} = 6 \text{ miles}$ - Walking: $3 \text{ mph} \times 0.5 \text{ hours} = 1.5 \text{ miles}$ Total distance = 6 + 1.5 = 7.5 miles Total time = 0.75 + 0.5 = 1.25 hours Average speed = $\frac{\text{total distance}}{\text{total time}} = \frac{7.5}{1.25} = 6.0 \text{ mph}$ Choice A (5.2 mph) likely results from incorrectly weighting the speeds by time rather than using the proper distance-time formula. Choice B (5.5 mph) is simply the arithmetic average of 8 mph and 3 mph: $\frac{8+3}{2} = 5.5$, which ignores the different time periods entirely. Choice C (5.8 mph) might come from a calculation error when finding the weighted average or making unit conversion mistakes. The key strategy here is to always find total distance and total time separately, then divide. Never just average the speeds unless the time periods are equal—this is a common trap on standardized tests.
13. Tom has 3 times as many baseball cards as soccer cards. He has 4 more football cards than soccer cards. If he has 84 cards total, how many soccer cards does he have?
    Answer: 16 soccer cards This is a classic algebra word problem that requires you to translate relationships between quantities into mathematical expressions. When you see phrases like "3 times as many" and "4 more than," you're dealing with a system where one unknown variable can help you find all the others. Let's call the number of soccer cards $s$. From the problem, Tom has $3s$ baseball cards and $s + 4$ football cards. Since the total is 84 cards, we can write: $s + 3s + (s + 4) = 84$. Simplifying: $5s + 4 = 84$, so $5s = 80$, which means $s = 16$ soccer cards. Let's verify: 16 soccer cards, $3 \times 16 = 48$ baseball cards, and $16 + 4 = 20$ football cards gives us $16 + 48 + 20 = 84$ total cards. ✓ Looking at the wrong answers: (A) 12 would give us $12 + 36 + 16 = 64$ total cards, falling short of 84. (B) 15 would yield $15 + 45 + 19 = 79$ cards, also too few. (D) 20 would result in $20 + 60 + 24 = 104$ cards, exceeding our target of 84. Strategy tip: In multi-step word problems, always define your variable clearly, translate each relationship into math, and check your answer by substituting back into the original conditions. This verification step catches calculation errors and confirms you've interpreted the relationships correctly.
14. A rectangular garden is 15 feet long and 8 feet wide. If a path 2 feet wide is built around the entire garden, what is the area of the path only?
    Answer: 108 square feet Find the area of the garden plus the path, then subtract the garden. The path adds 2 feet on every side, so each dimension grows by $2 + 2 = 4$ feet. The outer rectangle is $15 + 4 = 19$ feet by $8 + 4 = 12$ feet, with area $19 \times 12 = 228$ square feet. The garden alone is $15 \times 8 = 120$ square feet. The path is $228 - 120 = 108$ square feet. An answer of 92 square feet comes from adding four strips, $2 \times (2 \times 15) + 2 \times (2 \times 8) = 60 + 32$, which leaves out the four 2-by-2 corner squares ($4 \times 4 = 16$ square feet). An answer of 84 square feet comes from widening the length by only 2 feet ($17 \times 12 - 120$). An answer of 120 square feet is the garden itself, not the path. For borders, always add the path width to both ends of each dimension.
15. Lisa saves $12 per week for the first 6 weeks, then increases her savings to $18 per week for the next 8 weeks. She then buys a gift that costs 2/3 of her total savings. How much money does she have left?
    Answer: $72 This is a multi-step word problem that tests your ability to break down a savings scenario and apply fraction operations. When you see problems involving accumulating money over different time periods, work systematically through each phase. First, calculate Lisa's total savings. In the first 6 weeks, she saves $12 per week: $6 \times 12 = 72$ dollars. In the next 8 weeks, she saves $18 per week: $8 \times 18 = 144$ dollars. Her total savings is $72 + 144 = 216$ dollars. Next, determine how much she spends. The gift costs $\frac{2}{3}$ of her total savings: $\frac{2}{3} \times 216 = 144$ dollars. Finally, subtract what she spent from her total savings: $216 - 144 = 72$ dollars remaining. Looking at the wrong answers: Choice B ($84) might result from incorrectly calculating $\frac{1}{2} \times 216 - 24$, mixing up the fraction. Choice C ($96) could come from calculating $216 \times \frac{2}{3} - 48$, perhaps from an arithmetic error in the gift cost calculation. Choice D ($108) might result from finding $216 \times \frac{1}{2}$, using the wrong fraction entirely. The correct answer is A) $72. Strategy tip: In multi-step money problems, always organize your work: (1) find total money accumulated, (2) calculate money spent or saved using any fractions given, (3) subtract to find what remains. Double-check your fraction arithmetic since these problems often hinge on that calculation.
16. Maria bought 3 notebooks for $4.50 each and some pens for $1.25 each. If she spent a total of $22.25, how many pens did she buy?
    Answer: 7 This is a classic two-step word problem that tests your ability to set up and solve equations with multiple purchases. When you see problems involving different items with different prices and a total cost, you'll need to account for each type of purchase separately. Start by calculating what Maria spent on notebooks: 3 notebooks × $4.50 each = $13.50. Since she spent $22.25 total, the amount spent on pens must be $22.25 - $13.50 = $8.75. Now divide the pen expenditure by the price per pen: $8.75 ÷ $1.25 = 7 pens. Let's check why the other answers are wrong. Choice B (8 pens) would cost 8 × $1.25 = $10.00, making her total spending $13.50 + $10.00 = $23.50, which exceeds her actual total. Choice C (9 pens) would cost 9 × $1.25 = $11.25, giving a total of $13.50 + $11.25 = $24.75, which is even higher than her actual spending. Choice D (10 pens) would cost 10 × $1.25 = $12.50, resulting in a total of $13.50 + $12.50 = $26.00, which is far too much. The correct answer is A (7 pens). For multi-item purchase problems, always organize your work systematically: calculate the cost of known quantities first, subtract from the total to find what's left for unknown quantities, then divide by the unit price. This step-by-step approach prevents calculation errors and makes checking your work easier.
17. A machine fills bottles at a rate of 45 bottles per minute for the first 8 minutes, then slows to 30 bottles per minute for the next 12 minutes. How many bottles are filled in total?
    Answer: 720 bottles When you encounter a rate problem with changing conditions, you need to calculate each phase separately and then add the results together. Let's break this into two distinct time periods. In the first 8 minutes, the machine fills 45 bottles per minute. So the total for this period is: $45 \text{ bottles/minute} \times 8 \text{ minutes} = 360 \text{ bottles}$ In the second phase, the machine slows down to 30 bottles per minute for 12 minutes. This gives us: $30 \text{ bottles/minute} \times 12 \text{ minutes} = 360 \text{ bottles}$ Adding both phases together: $360 + 360 = 720 \text{ bottles}$ Looking at the wrong answers, choice A (360 bottles) represents only the first 8 minutes of production—a common trap where students calculate just one phase and stop. Choice B (540 bottles) might result from incorrectly using 45 bottles/minute for only 12 minutes, mixing up the rates and time periods. Choice D (900 bottles) could come from using the faster rate (45 bottles/minute) for the entire 20-minute period, ignoring the fact that the machine slowed down. The key strategy for multi-step rate problems is to organize your work clearly: identify each distinct phase, calculate each one separately using the rate formula (rate × time = quantity), then combine your results. Always double-check that you've accounted for all time periods and used the correct rate for each phase.
18. A water tank is initially 3/4 full of its 240-gallon capacity. Due to a leak, it loses 1/3 of its current water overnight. The next day, 45 gallons are added. What fraction of the tank's capacity is now filled?
    Answer: 11/16 When you encounter multi-step fraction problems involving capacity, work through each change systematically and keep track of the actual amounts rather than jumping straight to fractions. Start with the initial amount: $\frac{3}{4} \times 240 = 180$ gallons. After losing $\frac{1}{3}$ of the current water overnight, the tank loses $\frac{1}{3} \times 180 = 60$ gallons, leaving $180 - 60 = 120$ gallons. Adding 45 gallons the next day gives us $120 + 45 = 165$ gallons total. To find what fraction this represents: $\frac{165}{240} = \frac{165 \div 15}{240 \div 15} = \frac{11}{16}$. Since this isn't among choices A-D, the correct answer must be E. Let's see where each wrong answer comes from. Choice A ($\frac{7}{16}$) equals 105 gallons—you'd get this if you incorrectly calculated the overnight loss. Choice B ($\frac{1}{2}$) equals 120 gallons—this is the amount before adding the 45 gallons, suggesting you forgot the final step. Choice C ($\frac{9}{16}$) equals 135 gallons—this might result from calculation errors in the loss amount. Choice D ($\frac{5}{8}$) equals 150 gallons, which doesn't correspond to any logical step in the problem. The key strategy here is recognizing that not all multiple-choice questions have the answer listed in the visible choices. When working with fractions and capacities, always convert to actual amounts first, complete all operations, then convert back to the fraction form. This prevents compounding errors from working with fractions throughout multiple steps.
19. A box contains red, blue, and green marbles in the ratio 3:4:5. If there are 15 green marbles, what is the total number of marbles in the box?
    Answer: 36 marbles When you encounter ratio problems, you're working with proportional relationships between different quantities. The key insight is that ratios tell you the relative sizes of groups, and you can use any one known quantity to find all the others. The ratio 3:4:5 means that for every 3 red marbles, there are 4 blue marbles and 5 green marbles. Since you know there are 15 green marbles, you can find the "multiplier" that scales the ratio to the actual quantities. If the ratio calls for 5 green marbles but you actually have 15, then $15 ÷ 5 = 3$, so the multiplier is 3. This means you have: - Red marbles: $3 × 3 = 9$ - Blue marbles: $4 × 3 = 12$ - Green marbles: $5 × 3 = 15$ ✓ Total marbles: $9 + 12 + 15 = 36$ Looking at the wrong answers: Choice A (30) likely comes from incorrectly thinking the ratio parts should add to 30 when scaled. Choice B (32) might result from calculation errors in the multiplication or addition. Choice C (35) could come from forgetting to multiply one of the ratio parts by 3, perhaps counting 5 green marbles instead of 15. The correct answer is D) 36 marbles. Strategy tip: In ratio problems, always identify what quantity you know, find the multiplier by dividing the known quantity by its ratio part, then apply that multiplier to all parts of the ratio. Double-check by verifying the known quantity matches what you're given.
20. A rectangular prism has dimensions 4 cm by 6 cm by 8 cm. If each dimension is increased by 2 cm, by how much does the volume increase?
    Answer: 288 cubic cm When you encounter volume problems involving dimensional changes, you need to calculate the original volume, find the new volume after the changes, then determine the difference. First, find the original volume of the rectangular prism. Volume equals length × width × height, so: $4 \times 6 \times 8 = 192$ cubic cm. Next, calculate the new dimensions after each is increased by 2 cm: the dimensions become 6 cm, 8 cm, and 10 cm. The new volume is: $6 \times 8 \times 10 = 480$ cubic cm. The volume increase is: $480 - 192 = 288$ cubic cm, which is answer choice C. Now let's examine why the other answers are incorrect. Choice A (144 cubic cm) represents exactly three-quarters of the original volume, which might result from incorrectly calculating $2 \times 6 \times 12$ or similar computational errors. Choice B (192 cubic cm) is actually the original volume itself—this trap catches students who calculate the new volume but forget to subtract the original volume. Choice D (336 cubic cm) could result from miscalculating the new dimensions or making arithmetic errors during multiplication. Strategy tip: For volume change problems, always follow the same three-step process: calculate original volume, calculate new volume, then find the difference. Double-check your arithmetic since these problems involve multiple multiplications where small errors compound quickly. Also, be wary of answer choices that equal the original volume—they're often included as traps.

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