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.

When you encounter problems about paths or borders around rectangular areas, you need to find the difference between the total area (garden plus path) and the original area.

Start by visualizing the situation: the original garden is 15 feet by 8 feet, and a 2-foot-wide path surrounds it completely. This means the path adds 2 feet on all four sides of the garden.

The new outer dimensions become: length = 15 + 2 + 2 = 19 feet, and width = 8 + 2 + 2 = 12 feet. The total area (garden plus path) is $$19 \times 12 = 228$$ square feet.

The original garden area is $$15 \times 8 = 120$$ square feet.

Therefore, the path area equals: $$228 - 120 = 108$$ square feet.

Wait - that's not among the choices! Let me reconsider the problem. Looking at the answer choices, choice C (92 square feet) suggests a different interpretation might be intended, but the calculation above is mathematically correct for a 2-foot path on all sides.

However, examining the choices: A) 76 could result from calculation errors with dimensions. B) 84 might come from using incorrect outer dimensions. D) 120 is exactly the garden's area - a common trap where students give the original area instead of the path area.

The discrepancy suggests there may be an error in the problem setup or answer choices, but C) 92 is marked correct.

Study tip: Always double-check path problems by ensuring you add the path width to both sides of each dimension, and remember that path area = total area minus original area.

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.

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.

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.

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.

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.

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.

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.

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.