Julian and Frank share a watermelon. Frank asks for \(\dfrac{10_{ten}}{21_{ten}}\) of the whole. Julian gives \(\dfrac{24_{eight}}{52_{eight}}\) of the whole. Did Julian give the correct amount? Explain.
In base twelve (with \(A=10\text{,}\)\(B=11\)), find four distinct fractions strictly between \(\dfrac{A_{twelve}}{10_{twelve}}\) and \(\dfrac{B_{twelve}}{10_{twelve}}\text{.}\)
Describe a different scenario (choose slice counts, original partitions, number of friends) where they can share equally with exactly one cut. Explain.
Create your own scenario described by a ratio between two quantities, and specify a numerical request that is impossible because of that ratio. Explain why it is impossible.
Store A (Bucket O’ Hammers) sells \(10\,\text{m}\) of fencing for \(200\)\(\$\text{.}\) Store B (Screws, Nails and Everything Lumber) sells \(15\,\text{m}\) for \(285\)\(\$\text{.}\) Pious doesn’t mind leftovers—he wants the best value. Which store should he choose?
Suppose instead he wants a rectangular pen of area \(40\,\text{m}^2\text{.}\) What is the minimum fencing required, and which store yields the lowest cost?
The next day his overall rate was \(\dfrac{10\ \text{baskets}}{\text{game}}\) over 4 games. He scored 7, 8, and 12 in three of them. How many in the fourth?
The payroll administrator at Bland Media Conglomerate LLC. reports wages and earnings in base twelve, while hours remain in base ten. Do not convert to base ten; work directly in the indicated bases.
After working 35 hours at \(\dfrac{20_{\text{twelve}}\ \text{\$}}{\text{hr}}\text{,}\) Julie received \(6A0_{\text{twelve}}\)\(\$\text{.}\) Was there an error? What should she have been paid?
Greg worked 40 regular hours and 10 overtime hours. Overtime is paid at twice his regular wage. He was paid \(10A2_{\text{twelve}}\)\(\$\text{.}\) What is his hourly rate (in base twelve)?
Jerry was paid \(1480_{\text{twelve}}\)\(\$\text{.}\) A fraction \(\dfrac{7_{\text{twelve}}}{A_{\text{twelve}}}\) of his hours were at regular pay and the remaining \(\dfrac{3_{\text{twelve}}}{A_{\text{twelve}}}\) were overtime. What is Jerry’s hourly wage (in base twelve)?
Fractions can mean different things in different contexts (part–whole, quotient, ratio, operator, measure). For each meaning, give a short real-world example for \(\tfrac{3}{4}\) and explain why that meaning fits.
A student says: “Since \(\tfrac{7}{12}\) and \(\tfrac{2}{3}\) both have even numerators and denominators, \(\tfrac{7}{12}<\tfrac{2}{3}\) because \(7<8\) and \(12<12\text{.}\)” Critique this reasoning carefully. Give a correct method to compare \(\tfrac{7}{12}\) and \(\tfrac{2}{3}\) and explain why it works.
(Mediant reasoning) Suppose \(\tfrac{a}{b}\) and \(\tfrac{c}{d}\) are positive fractions with \(\tfrac{a}{b}<\tfrac{c}{d}\text{.}\) Explain why the fraction \(\tfrac{a+c}{\,b+d\,}\) lies strictly between them. State any conditions you use (e.g., positivity) and justify in words and/or symbols.
Explain why there are infinitely many fractions between \(\tfrac{1}{3}\) and \(\tfrac{1}{2}\text{.}\) Give a method that generates different examples on demand and explain why it always stays in the interval.
(Base-independence) A rule of simplification says: if \(k\neq0\text{,}\) then \(\tfrac{ak}{bk}=\tfrac{a}{b}\text{.}\) Explain why this statement is true regardless of the base used to write the digits of \(a,b,k\text{.}\) Your explanation should separate “the quantities” from “their written representations.”
Identify the exact step where a property is misused. Then show two correct explanations for adding these fractions: (i) common denominator, (ii) visual model (describe the picture you would draw).
Explain why the rule “to convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator” is correct. Your explanation should work for an arbitrary \(W+\tfrac{N}{D}\text{,}\) and may use a diagram or algebraic reasoning.
Multiplying fractions: Without doing any simplification first, explain why \(\tfrac{2}{3}\times\tfrac{3}{5}=\tfrac{2}{5}\) using an area model. Describe clearly how each factor is represented and why the overlapping region has the stated fraction of the whole.
Dividing by a fraction: Explain, using a measurement or sharing interpretation, why \(\tfrac{3}{4}\div\tfrac{2}{5}=\tfrac{15}{8}\text{.}\) Your explanation should make it clear why “multiply by the reciprocal” matches the story.
Cross-multiplication for comparison: To decide whether \(\tfrac{7}{11}\) is greater than \(\tfrac{5}{8}\text{,}\) one method forms \(7\cdot8\) and \(5\cdot11\text{.}\) Explain why this method is valid and how it relates to rewriting both fractions with a common denominator without actually finding the common denominator first.
(Betweeness via common denominators) Explain why, once two positive fractions are expressed with a common positive denominator, the one with the larger numerator is the larger fraction. Include a number-line interpretation that works even when the denominators are large.
(Fractions across bases) Explain why the statement \(\dfrac{10_{\text{four}}}{20_{\text{four}}}=\dfrac{1}{2}\) is correct without converting any numbers into base ten. Describe the quantities each numeral represents in base four and how the equality follows from those quantities.
Least common denominator (LCD): A student always chooses the product of denominators as the common denominator. Explain when this is unnecessarily large, how to find a smaller LCD using prime factors, and why using the LCD can make computation and simplification clearer.
(Equivalent-fraction generation) Give a principled way to generate all fractions equivalent to \(\tfrac{9}{12}\text{.}\) Explain why every fraction produced by your method is equivalent, and why any equivalent fraction must arise from your method.
(Order of operations with fractions) Consider \(\tfrac{3}{4}+\tfrac{1}{4}\times\tfrac{7}{9}\text{.}\) Explain two correct solution paths—one emphasizing order of operations first, and one rewriting everything over a common denominator—and justify why both lead to the same result.
(Reason about size) Without computing exact values, determine whether each expression is less than, equal to, or greater than \(1\text{,}\) and explain your reasoning.
(Complex fractions) Explain a clear strategy for simplifying \(\dfrac{\tfrac{3}{5}+\tfrac{1}{2}}{\tfrac{4}{3}-\tfrac{1}{6}}\) that avoids arithmetic mistakes. Justify why your strategy works and describe how you would check your final answer for reasonableness.
