otation?","back":"Dots are placed over the first and last digits of the repeating block.","points":1},{"id":"753bd71a-c67a-4dc7-a48a-649275f7c372","cardType":"cloze","front":"In the notation $0.2\\dot{4}\\dot{9}$, only the digits [ ] repeat.","back":"In the notation $0.2\\dot{4}\\dot{9}$, only the digits 4 and 9 repeat.","points":null},{"id":"901cb2f6-4f6a-45f4-9699-9371b7742cf6","cardType":"basic","front":"When converting a fraction using long division, how do you know the decimal will repeat?","back":"When the same remainder appears for a second time.","points":2},{"id":"0955ab14-5dfe-41b3-8979-49a6cf956e7d","cardType":"basic","front":"What is the core purpose of multiplying a recurring decimal by powers of 10 in algebra?","back":"To create two equations where the recurring tails are identical.","points":2},{"id":"9d548039-5314-46e8-a4b0-afbe6e9cf0c2","cardType":"basic","front":"Why is subtraction used after lining up the recurring tails of two equations?","back":"To cancel out the infinite repeating part of the decimal.","points":2},{"id":"b41a1f64-18c1-4764-9fd9-073c4ae0ea2b","cardType":"basic","front":"Outline the algebraic steps to prove $0.\\dot{6} = \\frac{2}{3}$.","back":"

Let $x = 0.666\\dots$
Multiply by 10 ($10x = 6.666\\dots$)
Subtract ($9x = 6$) and simplify $x = \\frac{6}{9}$

","points":3},{"id":"e8276a4d-d226-4431-8649-4c6a9448888d","cardType":"basic","front":"What constitutes a fraction in its simplest form?","back":"The numerator and denominator have no common factors other than 1.","points":1},{"id":"ef935412-6d9d-4177-b09c-de7fc0b96acf","cardType":"basic","front":"To convert $0.3\\dot{8}$ to a fraction, which two multiples of $x$ should be compared to align the tails?","back":"$$100x$ and $10x$.","points":1},{"id":"11c412e6-8c31-47c8-be73-5fb0e873087c","cardType":"cloze","front":"To align tails for a decimal with $n$ non-recurring digits and a block of $r$ recurring digits, compare $\\text{[...]}$ and $10^n x$.","back":"To align tails for a decimal with $n$ non-recurring digits and a block of $r$ recurring digits, compare $\\boxed{10^{n+r}x}$ and $10^n x$.","points":null},{"id":"9789f7df-3cea-40a7-b512-f6cde454de08","cardType":"cloze","front":"To align tails for a decimal with $n$ non-recurring digits and a block of $r$ recurring digits, compare $10^{n+r}x$ and $\\text{[...]}$.","back":"To align tails for a decimal with $n$ non-recurring digits and a block of $r$ recurring digits, compare $10^{n+r}x$ and $\\boxed{10^n x}$.","points":null},{"id":"d7d547d8-ee04-47a6-830f-ce3d3188b9b6","cardType":"basic","front":"For the decimal $0.\\dot{4}\\dot{7}$, why is the decimal multiplied by 100 instead of 10?","back":"The repeating block has 2 digits; multiplying by 100 shifts the tail to match.","points":2},{"id":"690a66ad-a2ae-48ad-af66-6c177fa4a630","cardType":"basic","front":"Is it necessary to separate the whole number (e.g., the 3 in $3.\\dot{2}$) before using the algebra trick?","back":"No; let $x$ equal the entire decimal and use the standard method.","points":1},{"id":"4ca3a126-4963-4537-b1a6-1a851ffb1c33","cardType":"cloze","front":"The final answer for a decimal greater than 1 (like $3.2\\dot{6}\\dot{3}$) is often an [ ].","back":"The final answer for a decimal greater than 1 (like $3.2\\dot{6}\\dot{3}$) is often an improper fraction.","points":null},{"id":"919770c2-46fa-421c-b2da-e2565c665f3a","cardType":"basic","front":"What is the resulting fraction for the recurring digit $a$ in the decimal $0.\\dot{a}$?","back":"$$\\frac{a}{9}$","points":1},{"id":"1bf9627f-428b-4915-b6ce-8f5ad4aa4d34","cardType":"basic","front":"What is the resulting fraction for the repeating block in $0.\\dot{0}\\dot{a}$?","back":"$$\\frac{a}{99}$

Recurring Decimals to Fractions Flashcards

Recurring Decimals to Fractions Flashcards

Recurring Decimals to Fractions Flashcards

Recurring Decimals to Fractions Flashcards