This question tests ISEE Upper Level Mathematics Achievement skills: add, subtract, multiply, and divide fractions. Fractions are numbers that represent parts of a whole. Operations with fractions involve adding, subtracting, multiplying, and dividing these numbers while adhering to rules specific to fraction arithmetic. In the given scenario, students must apply these rules to solve a problem involving subtracting 5/6 L from 2 1/3 L of liquid in a beaker. The correct answer is choice A, 1 1/2 L, which accurately represents the result of converting to common denominators (2 1/3 = 2 2/6, then 2 2/6 - 5/6 = 1 3/6 = 1 1/2). Choice B (3 1/6) is incorrect due to adding instead of subtracting, a common mistake when students misread the operation. To assist students, encourage the practice of converting fractions to common denominators for addition and subtraction, and using reciprocal operations for division.

When you encounter a recipe problem involving fractions, you need to scale the ingredients proportionally and then combine them as requested.

First, let's find how much of each ingredient Maria needs for half the recipe. For flour: $$\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$$ cup. For sugar: $$\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}$$ cup.

Now you need to add these fractions: $$\frac{3}{8} + \frac{1}{3}$$. To add fractions, find a common denominator. The LCD of 8 and 3 is 24. Convert each fraction: $$\frac{3}{8} = \frac{9}{24}$$ and $$\frac{1}{3} = \frac{8}{24}$$. Therefore: $$\frac{9}{24} + \frac{8}{24} = \frac{17}{24}$$ cups total.

Choice A ($$\frac{17}{24}$$) is correct. Choice B ($$\frac{5}{6}$$) represents a common error of adding the original amounts first ($$\frac{3}{4} + \frac{2}{3} = \frac{17}{12}$$) then halving incorrectly. Choice C ($$\frac{7}{12}$$) occurs if you mistakenly use 12 as your common denominator when adding $$\frac{3}{8}$$ and $$\frac{1}{3}$$. Choice D ($$\frac{11}{24}$$) results from calculation errors in the addition step.

Remember: in multi-step fraction problems, work systematically—scale each ingredient separately first, then perform the final operation. Always double-check your common denominator, as LCD errors are frequent traps on fraction problems.

This problem tests your ability to work with fractions of fractions — a key skill where you need to carefully track what each fraction refers to.

Start by identifying what you know: the tank is $$\frac{7}{8}$$ full, and you use $$\frac{1}{3}$$ of the water currently in the tank. The key insight is that $$\frac{1}{3}$$ refers to one-third of the water that's actually there, not one-third of the tank's total capacity.

To find how much water is used, multiply: $$\frac{1}{3} \times \frac{7}{8} = \frac{7}{24}$$ of the tank's total capacity is removed.

The remaining water is: $$\frac{7}{8} - \frac{7}{24}$$. To subtract these fractions, find a common denominator of 24: $$\frac{21}{24} - \frac{7}{24} = \frac{14}{24} = \frac{7}{12}$$.

Looking at the wrong answers: Choice B ($$\frac{5}{8}$$) comes from incorrectly thinking you remove $$\frac{1}{3}$$ from $$\frac{7}{8}$$ by subtracting the numerators: $$7-1=6$$, then simplifying $$\frac{6}{8}$$ to $$\frac{3}{4}$$ — but this isn't among the choices, leading to further errors. Choice C ($$\frac{13}{24}$$) results from adding instead of subtracting during the calculation. Choice D ($$\frac{7}{24}$$) is the amount of water removed, not the amount remaining.

When you see "fraction of fraction" problems, always identify what the second fraction refers to — is it a fraction of the total capacity or a fraction of what's currently there? This distinction is crucial for setting up your calculation correctly.

When you see mixed operations with fractions, remember that order of operations (PEMDAS) still applies—multiplication comes before addition, even with fractions.

First, handle the multiplication: $$\frac{5}{8} \times \frac{2}{15}$$. Multiply the numerators together and denominators together: $$\frac{5 \times 2}{8 \times 15} = \frac{10}{120}$$. Simplify by dividing both parts by their greatest common factor of 10: $$\frac{10}{120} = \frac{1}{12}$$.

Now add: $$\frac{2}{3} + \frac{1}{12}$$. To add fractions, you need a common denominator. The least common multiple of 3 and 12 is 12. Convert $$\frac{2}{3}$$ to twelfths: $$\frac{2}{3} = \frac{8}{12}$$. Therefore: $$\frac{8}{12} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4}$$, which is choice B.

Choice A ($$\frac{17}{24}$$) likely comes from using 24 as a common denominator incorrectly or making computational errors. Choice C ($$\frac{13}{24}$$) might result from adding all three fractions as if they were separate terms, ignoring the multiplication. Choice D ($$\frac{11}{24}$$) could come from similar computational mistakes with the wrong denominator.

Strategy tip: Always follow order of operations with fractions just like with whole numbers. Complete all multiplication and division first, then handle addition and subtraction. Double-check by simplifying fractions at each step—it makes the final computation much easier.

When you see fractions being multiplied together, you can use the fundamental property that multiplying fractions means multiplying numerators together and denominators together: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$.

Since you're given that $$\frac{a}{b} = \frac{3}{7}$$ and $$\frac{c}{d} = \frac{2}{5}$$, you can substitute these values directly into the multiplication: $$\frac{a \cdot c}{b \cdot d} = \frac{3}{7} \cdot \frac{2}{5} = \frac{3 \cdot 2}{7 \cdot 5} = \frac{6}{35}$$. This matches choice A.

Let's examine why the other answers are incorrect. Choice B gives $$\frac{5}{12}$$, which might result from incorrectly adding the fractions instead of multiplying them, or from some other computational error. Choice C shows $$\frac{1}{6}$$, which doesn't follow from any reasonable operation on the given fractions. Choice D presents $$\frac{21}{10}$$, which you might get if you mistakenly flipped one of the fractions or cross-multiplied incorrectly.

The key insight here is recognizing that $$\frac{a \cdot c}{b \cdot d}$$ is simply asking for the product of two fractions you already know. Remember that when multiplying fractions, you multiply straight across - numerator times numerator, denominator times denominator. Don't overthink these problems by trying to find individual values for a, b, c, and d when the given ratios are sufficient to solve directly.

This question tests your ability to work with fraction division and reciprocals in sequence. When you see a complex fraction expression followed by "find the reciprocal," break it down step by step rather than trying to do everything at once.

First, you need to evaluate $$\frac{2}{3} \div \frac{4}{9}$$. Remember that dividing by a fraction means multiplying by its reciprocal: $$\frac{2}{3} \div \frac{4}{9} = \frac{2}{3} \times \frac{9}{4}$$. Multiplying across gives you $$\frac{2 \times 9}{3 \times 4} = \frac{18}{12}$$. This simplifies to $$\frac{3}{2}$$ when you divide both numerator and denominator by 6.

Now you need the reciprocal of $$\frac{3}{2}$$, which is $$\frac{2}{3}$$. This makes choice A correct.

Choice B ($$\frac{3}{2}$$) is the result of the division but not its reciprocal - a common error when students forget the final step. Choice C ($$\frac{1}{2}$$) likely comes from incorrectly calculating the original division, possibly by multiplying $$\frac{2}{3} \times \frac{4}{9}$$ instead of dividing. Choice D ($$\frac{9}{8}$$) might result from flipping the wrong fraction during the division step or making arithmetic errors in the multiplication.

The key strategy here is to work systematically: first complete the division operation, then find the reciprocal of that result. Don't try to take shortcuts by looking for reciprocals earlier in the process, as this often leads to confusion about which fraction to flip.

When you encounter a word problem involving division of fractions, you're essentially finding how many times one quantity fits into another, plus any remainder.

To find how many complete pieces you can make, divide the total rope length by the length of each piece: $$\frac{7}{8} ÷ \frac{1}{6}$$. When dividing fractions, multiply by the reciprocal: $$\frac{7}{8} × \frac{6}{1} = \frac{42}{8} = \frac{21}{4} = 5\frac{1}{4}$$. Since you can only make complete pieces, you get 5 pieces.

To find the remaining rope, multiply the number of complete pieces by the piece length, then subtract from the original length: $$5 × \frac{1}{6} = \frac{5}{6}$$. Now calculate $$\frac{7}{8} - \frac{5}{6}$$. Using the common denominator 24: $$\frac{21}{24} - \frac{20}{24} = \frac{1}{24}$$. Therefore, answer A is correct.

Choice B incorrectly assumes only 4 complete pieces can be made, which underestimates the division result. Choice C correctly identifies 5 pieces but miscalculates the remainder—this likely comes from arithmetic errors in fraction subtraction. Choice D combines both errors: wrong number of pieces and wrong remainder calculation.

Remember that division word problems with fractions have two parts: the whole number tells you complete units, while the fractional part helps you calculate the remainder. Always convert mixed numbers properly and double-check your fraction arithmetic using common denominators.

When you encounter a complex fraction (a fraction containing other fractions), your goal is to simplify both the numerator and denominator separately, then divide.

Let's start with the numerator: $$\frac{3}{4} + \frac{1}{8}$$. To add fractions, you need a common denominator. Since 8 is a multiple of 4, convert $$\frac{3}{4}$$ to eighths: $$\frac{3}{4} = \frac{6}{8}$$. Now add: $$\frac{6}{8} + \frac{1}{8} = \frac{7}{8}$$.

For the denominator: $$\frac{5}{6} - \frac{1}{3}$$. The common denominator is 6, so convert $$\frac{1}{3}$$ to sixths: $$\frac{1}{3} = \frac{2}{6}$$. Now subtract: $$\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$.

Your complex fraction is now $$\frac{\frac{7}{8}}{\frac{1}{2}}$$. To divide by a fraction, multiply by its reciprocal: $$\frac{7}{8} \times \frac{2}{1} = \frac{14}{8} = \frac{7}{4}$$. This confirms answer A.

Answer B ($$\frac{21}{8}$$) results from incorrectly multiplying the numerator by 3 instead of properly finding the common denominator. Answer C ($$\frac{5}{3}$$) comes from calculation errors in both the numerator and denominator operations. Answer D ($$\frac{7}{2}$$) occurs when you forget to reduce $$\frac{3}{6}$$ to $$\frac{1}{2}$$ in the denominator, leading to $$\frac{7}{8} \div \frac{3}{6} = \frac{7}{8} \times \frac{6}{3} = \frac{7}{4} \times \frac{3}{2}$$.

Remember: always simplify fractions completely at each step, and double-check your common denominators before adding or subtracting.

This problem tests your ability to work with exponents and division of fractions. When you see division by a fraction, remember that dividing by a fraction is the same as multiplying by its reciprocal.

First, calculate $$\left(\frac{2}{3}\right)^2$$. When you square a fraction, you square both the numerator and denominator: $$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$$.

Now you need to compute $$\frac{4}{9} \div \frac{8}{27}$$. To divide by a fraction, multiply by its reciprocal: $$\frac{4}{9} \times \frac{27}{8}$$.

Multiply the fractions: $$\frac{4 \times 27}{9 \times 8} = \frac{108}{72}$$.

Simplify by finding the greatest common factor. Both 108 and 72 are divisible by 36: $$\frac{108}{72} = \frac{108 \div 36}{72 \div 36} = \frac{3}{2}$$.

Looking at the wrong answers: Choice A ($$\frac{1}{2}$$) likely results from incorrectly multiplying instead of dividing, or making arithmetic errors. Choice C ($$\frac{2}{3}$$) might come from confusing the original fraction with the final answer. Choice D ($$\frac{4}{9}$$) is what you get if you stop after squaring $$\frac{2}{3}$$ and forget to complete the division.

The correct answer is B: $$\frac{3}{2}$$.

Strategy tip: When dividing fractions, always convert to multiplication by the reciprocal immediately. This prevents confusion and makes the arithmetic cleaner. Also, look for opportunities to simplify before multiplying to make calculations easier.

This is a proportion problem where you need to scale a recipe up from 8 people to 12 people. When scaling recipes, you're looking for the relationship between the original serving size and the new serving size, then applying that same ratio to each ingredient.

First, find the scaling factor: $$\frac{12 \text{ people}}{8 \text{ people}} = \frac{12}{8} = \frac{3}{2}$$

This means you need 1.5 times the original recipe. Now multiply the flour amount by this factor:

$$\frac{5}{6} \times \frac{3}{2} = \frac{5 \times 3}{6 \times 2} = \frac{15}{12}$$

Simplify this fraction: $$\frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}$$ cups

Choice A ($$1\frac{1}{4}$$ cups) is correct.

Choice B ($$1\frac{1}{2}$$ cups) likely comes from incorrectly calculating the scaling factor as $$\frac{12}{8} = 1.5$$ and then adding this to the original amount instead of multiplying. Choice C ($$1\frac{1}{3}$$ cups) might result from computational errors when multiplying fractions or confusion about mixed number conversion. Choice D ($$1\frac{1}{6}$$ cups) could come from adding $$\frac{5}{6} + \frac{2}{6}$$ (thinking you need $$\frac{2}{6}$$ more), which shows a misunderstanding of proportional scaling.

Remember: In proportion problems, always set up the ratio first (new amount ÷ original amount), then multiply each ingredient by this scaling factor. Don't add amounts—you're scaling the entire recipe proportionally.