When you encounter problems involving fractions of a whole group, remember that all the parts must add up to 1 (or 100%). The key is converting everything to the same format before calculating.

First, convert the decimal to a fraction: $$0.375 = \frac{375}{1000} = \frac{3}{8}$$ (dividing both numerator and denominator by 125).

Now you have $$\frac{3}{8}$$ prefer pizza and $$\frac{2}{5}$$ prefer burgers. To subtract these from 1, you need a common denominator. The least common multiple of 8 and 5 is 40.

Converting: $$\frac{3}{8} = \frac{15}{40}$$ and $$\frac{2}{5} = \frac{16}{40}$$

The fraction preferring salad is: $$1 - \frac{15}{40} - \frac{16}{40} = \frac{40}{40} - \frac{31}{40} = \frac{9}{40}$$

Choice B, $$\frac{9}{40}$$, is correct.

Choice A ($$\frac{1}{4} = \frac{10}{40}$$) is too large and likely comes from incorrectly converting 0.375 or making an arithmetic error. Choice C ($$\frac{7}{40}$$) suggests subtracting incorrectly, possibly computing $$\frac{16}{40} - \frac{15}{40}$$ instead of subtracting both from 1. Choice D ($$\frac{3}{20} = \frac{6}{40}$$) likely results from computational errors with the common denominator or fraction conversions.

Strategy tip: Always verify your fractions add to 1 as a check. Here: $$\frac{15}{40} + \frac{16}{40} + \frac{9}{40} = \frac{40}{40} = 1$$ ✓

When you encounter decimal-to-fraction problems, start by converting the decimal to its simplest fractional form, then work with the given conditions.

The decimal $$0.45$$ equals $$\frac{45}{100}$$. To reduce this to lowest terms, find the greatest common divisor of 45 and 100. Since $$45 = 9 \times 5$$ and $$100 = 20 \times 5$$, the GCD is 5. Therefore: $$\frac{45}{100} = \frac{9}{20}$$.

The original fraction in lowest terms is $$\frac{9}{20}$$. When the denominator increases by 5, the new fraction becomes $$\frac{9}{25}$$. Converting to decimal: $$\frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} = 0.36$$.

Looking at the wrong answers: Choice B (0.40) might tempt you if you incorrectly calculated $$\frac{9}{25}$$ or confused it with $$\frac{2}{5}$$. Choice C (0.42) could result from arithmetic errors in the conversion process. Choice D (0.50) represents $$\frac{1}{2}$$, which you might get if you mistakenly thought the original fraction was $$\frac{1}{2}$$ instead of properly reducing $$\frac{45}{100}$$.

The correct answer is A) 0.36.

Strategy tip: Always reduce fractions to lowest terms first before applying any changes. When converting decimals to fractions, look for common factors immediately—this prevents errors and makes subsequent calculations cleaner. Practice converting common decimals like 0.25, 0.45, and 0.75 to fractions until it becomes automatic.

When you encounter percentage problems that set two expressions equal to each other, you're looking at an equation that you can solve algebraically. The key is to translate the words into mathematical expressions and then solve for the unknown.

Let's translate this step by step. "$$x%$$ of 80" means $$\frac{x}{100} \times 80$$, and "$$\frac{3}{5}$$ of 40" means $$\frac{3}{5} \times 40$$. Setting them equal: $$\frac{x}{100} \times 80 = \frac{3}{5} \times 40$$.

First, simplify the right side: $$\frac{3}{5} \times 40 = \frac{3 \times 40}{5} = \frac{120}{5} = 24$$.

Now solve: $$\frac{x}{100} \times 80 = 24$$. Simplify the left side: $$\frac{80x}{100} = \frac{4x}{5} = 24$$. Multiply both sides by $$\frac{5}{4}$$: $$x = 24 \times \frac{5}{4} = \frac{120}{4} = 30$$.

Since 30 is a whole number, it's already in mixed number form.

Looking at the wrong answers: Choice B ($$24\frac{1}{2}$$) is the trap of stopping at the value of the right side of the equation without solving for $$x$$. Choice C ($$30\frac{1}{3}$$) might result from calculation errors in the fraction arithmetic. Choice D ($$32\frac{1}{4}$$) could come from incorrectly setting up the initial equation or making arithmetic mistakes during solving.

The answer is A.

Strategy tip: Always translate percentage problems into equations first, then solve systematically. Double-check by substituting your answer back into the original problem to verify it works.

When you encounter a repeating decimal like $$0.8\overline{3}$$, you need to convert it to a fraction using algebraic manipulation. The bar over the 3 means that digit repeats infinitely: $$0.8333...$$

Let $$x = 0.8\overline{3} = 0.8333...$$ To eliminate the repeating portion, multiply both sides by 10 (since one digit repeats): $$10x = 8.333...$$ Now subtract the original equation: $$10x - x = 8.333... - 0.833...$$ This gives you $$9x = 7.5$$, so $$x = \frac{7.5}{9} = \frac{75}{90}$$

To reduce to lowest terms, find the greatest common divisor of 75 and 90. Both are divisible by 15: $$\frac{75}{90} = \frac{75 ÷ 15}{90 ÷ 15} = \frac{5}{6}$$

Answer A, $$\frac{5}{6}$$, is correct. Answer B, $$\frac{25}{30}$$, reduces to $$\frac{5}{6}$$ but isn't in lowest terms as requested. Answer C, $$\frac{83}{100} = 0.83$$, represents the misconception of treating $$0.8\overline{3}$$ as simply $$0.83$$ without accounting for the infinite repetition. Answer D, $$\frac{75}{90}$$, is the unreduced form of the correct fraction—you correctly set up the equation but forgot to simplify.

Strategy tip: For repeating decimals, always multiply by the appropriate power of 10 based on how many digits repeat (10 for one digit, 100 for two digits, etc.), then subtract to eliminate the repetition. Don't forget to reduce your final fraction to lowest terms.

When you encounter percentage problems involving subgroups, you need to find what portion of the entire population meets multiple criteria. Here, you're looking for students who are both boys AND play soccer.

Start with what you know: 40% of the class are boys, and $$\frac{3}{8}$$ of those boys play soccer. To find the percentage of the entire class that consists of boys who play soccer, multiply these two fractions together.

First, convert 40% to a fraction: $$40% = \frac{40}{100} = \frac{2}{5}$$

Now multiply: $$\frac{2}{5} \times \frac{3}{8} = \frac{6}{40} = \frac{3}{20}$$

Convert back to a percentage: $$\frac{3}{20} = 0.15 = 15%$$

This confirms answer choice B is correct.

Looking at the wrong answers: Choice A (12%) likely comes from incorrectly calculating $$\frac{2}{5} \times \frac{3}{8}$$ or making an arithmetic error. Choice C (18%) might result from adding instead of multiplying the percentages, or miscalculating the fraction multiplication. Choice D (20%) could come from mistakenly using $$\frac{1}{2}$$ instead of $$\frac{2}{5}$$ for the percentage of boys, then multiplying $$\frac{1}{2} \times \frac{3}{8} = \frac{3}{16} ≈ 18.75%$$, then rounding incorrectly.

Remember: When finding the percentage of a total population that meets multiple criteria, multiply the individual percentages or fractions. Don't add them—that would give you something meaningless like "boys plus soccer players," not "boys who play soccer."

When you encounter fraction-to-percentage conversions, remember that converting a fraction to a percentage means finding what the fraction equals out of 100. The given information that $$\frac{2}{3} = 66\frac{2}{3}%$$ provides a helpful reference point.

To find what percent $$\frac{5}{6}$$ equals, multiply the fraction by 100: $$\frac{5}{6} \times 100 = \frac{500}{6}$$. Now divide: $$500 \div 6 = 83\frac{2}{6}$$. Since $$\frac{2}{6} = \frac{1}{3}$$, we get $$83\frac{1}{3}%$$.

You can verify this makes sense by comparing it to the given information. Since $$\frac{5}{6} = \frac{10}{12}$$ and $$\frac{2}{3} = \frac{8}{12}$$, we know $$\frac{5}{6}$$ is larger than $$\frac{2}{3}$$, so its percentage should be greater than $$66\frac{2}{3}%$$.

Choice A gives us $$83\frac{1}{3}%$$, which matches our calculation perfectly. Choice B ($$82\frac{1}{2}%$$) results from incorrectly calculating $$\frac{500}{6}$$ or making an arithmetic error in the division. Choice C ($$85\frac{5}{6}%$$) comes from the common mistake of not simplifying $$\frac{500}{6}$$ properly and leaving the fractional part as $$\frac{5}{6}$$ instead of converting to the mixed number correctly. Choice D ($$80\frac{2}{3}%$$) likely results from calculation errors in the multiplication or division steps.

Study tip: When converting fractions to percentages, always multiply by 100 first, then simplify your result completely. Double-check by ensuring your answer makes logical sense compared to benchmark fractions you know.

When you encounter a complex fraction like this, your goal is to simplify it step by step by converting all numbers to the same form—either all decimals or all fractions—then performing the operations.

Let's convert everything to fractions for easier calculation. First, convert the decimals: $$0.75 = \frac{3}{4}$$ and $$1.25 = \frac{5}{4}$$. Now our expression becomes $$\frac{\frac{3}{4} + \frac{1}{8}}{\frac{5}{4}}$$.

To add the fractions in the numerator, find a common denominator. Since $$\frac{3}{4} = \frac{6}{8}$$, we have $$\frac{6}{8} + \frac{1}{8} = \frac{7}{8}$$. Our expression is now $$\frac{\frac{7}{8}}{\frac{5}{4}}$$.

To divide by a fraction, multiply by its reciprocal: $$\frac{7}{8} \times \frac{4}{5} = \frac{28}{40} = \frac{7}{10}$$. This confirms answer choice A is correct.

Looking at the wrong answers: B) $$\frac{3}{4}$$ might result from incorrectly using just the $$0.75$$ without adding $$\frac{1}{8}$$. C) $$\frac{4}{5}$$ could come from forgetting to divide by $$1.25$$ entirely. D) $$\frac{5}{6}$$ might result from calculation errors when finding common denominators or performing the division.

For complex fraction problems on the ISEE, always convert mixed decimal-fraction expressions to a single form first. Work systematically through each step rather than trying mental shortcuts, as the test makers often include trap answers that correspond to common computational mistakes.

When you encounter ratio problems with a given total, you're working with proportional relationships where each part represents a fraction of the whole. The key is to find what one "unit" of the ratio equals, then calculate each ingredient amount.

The ratio $$3:4:5$$ means the ingredients are in proportional amounts of 3 parts, 4 parts, and 5 parts respectively. First, find the total number of ratio parts: $$3 + 4 + 5 = 12$$ parts total.

Since the total amount is 2.4 cups, each part equals $$\frac{2.4}{12} = 0.2$$ cups per part.

The second ingredient needs 4 parts, so: $$4 \times 0.2 = 0.8$$ cups.

Looking at the wrong answers: Choice (A) 0.6 cups would result from incorrectly using 3 parts instead of 4 parts for the second ingredient ($$3 \times 0.2 = 0.6$$). Choice (C) 1.0 cups represents 5 parts ($$5 \times 0.2 = 1.0$$), which would be the third ingredient, not the second. Choice (D) 1.2 cups doesn't correspond to any single ingredient in this ratio—it would require 6 parts, which exceeds any individual component.

The correct answer is (B) 0.8 cups.

Strategy tip: Always set up ratio problems by first finding the total number of parts, then determining the value of one part by dividing the given total by this sum. This systematic approach prevents mix-ups between different ratio components and ensures accurate calculations.

When you encounter a problem where a fraction equals a percentage, you're dealing with part-to-whole relationships. The key insight is that the fraction $$\frac{17}{20}$$ and 85% both represent the same ratio of correct answers to total questions.

To solve this, convert the percentage to a fraction: $$85% = \frac{85}{100} = \frac{17}{20}$$. Since we're told that $$\frac{17}{20}$$ of the questions were answered correctly and this equals 85%, we can set up the equation: $$\frac{17}{20} = \frac{85}{100}$$. Cross-multiplying: $$17 \times 100 = 85 \times 20$$, which gives us $$1700 = 1700$$. This confirms our fractions are equivalent.

Looking at $$\frac{17}{20}$$, we can see that 17 represents the number of correct answers and 20 represents the total number of questions on the test.

Choice A (17) represents only the number of correct answers, not the total questions. Choice C (23) might come from incorrectly adding 17 + 6, perhaps thinking you need to add something to get from the numerator to the total. Choice D (25) could result from the misconception that since $$\frac{17}{20} = 0.85$$, you might think 17 out of 25 gives 85% (but $$\frac{17}{25} = 0.68 = 68%$$).

Choice B (20) is correct because the denominator of $$\frac{17}{20}$$ directly tells us the total number of questions.

Strategy tip: When a fraction equals a given percentage, convert the percentage to its simplest fraction form. The denominator of that simplified fraction often reveals the total you're looking for.

When you encounter percentage depreciation problems, you're dealing with finding a remaining value after a percentage decrease. The key insight is that if something depreciates by a certain percentage, you keep the remaining percentage of the original value.

First, convert the mixed number percentage to a decimal. $$16\frac{2}{3}% = 16.667% = \frac{50}{3}% = \frac{1}{6}$$ of the original value. If the car depreciates by $$\frac{1}{6}$$, then it retains $$1 - \frac{1}{6} = \frac{5}{6}$$ of its original value.

Calculate the remaining value: $$\frac{5}{6} \times \18,000 = \frac{5 \times 18,000}{6} = \frac{90,000}{6} = \15,000$$

Choice A ($$\14,800$$) likely comes from incorrectly calculating $$16\frac{2}{3}%$$ as $$17.78%$$ and subtracting $$\3,200$$ instead of $$\3,000$$. Choice C ($$\15,200$$) suggests someone calculated a $$15.56%$$ depreciation ($$\2,800$$ loss) instead of $$16\frac{2}{3}%$$. Choice D ($$\15,600$$) represents a $$13.33%$$ depreciation ($$\2,400$$ loss), possibly from misreading the fraction.

Remember that $$16\frac{2}{3}% = \frac{1}{6}$$ is a common fraction that appears frequently on standardized tests. When you see this percentage, immediately think "one-sixth," which makes the mental math much easier. Always double-check that you're subtracting the depreciation from 100% to find what percentage remains, then multiply by the original value.