\\frac{10}{13} = 0.\overline{769230}\

15. in the infinitely repeating decimal above, 7 is the first digit in the repeating pattern. what is the $391^{\text{st}}$ digit?
(a) 0
(b) 3
(c) 6
(d) 7

16. bryan completed a 100-meter race in 11.74 seconds. luis completed the same race in 11.69 seconds. what was the difference between their times, expressed as a fraction of a second?
(a) $\frac{1}{20}$
(b) $\frac{1}{10}$
(c) $\frac{3}{20}$
(d) $\frac{1}{5}$

\6.3\overline{4} - 4.1\overline{2}\

17. what is the value of the expression above, written as an improper fraction?
(a) $\frac{24}{11}$
(b) $\frac{224}{101}$
(c) $\frac{111}{50}$
(d) $\frac{20}{9}$

18. the decimal $0.0\overline{6}$ can be written as the fraction $\frac{x}{90}$. what is the value of $x$?
(a) 3
(b) 6
(c) 12
(d) 30

19. what decimal is equivalent to $-\frac{13}{11}$?
(a) $-1.18$
(b) $-1.118$
(c) $-1.\overline{18}$
(d) $-1.\overline{81}$

20. the decimal $0.4$ can be written as the fraction $\frac{x}{25}$. what is the value of $x$?
(a) 0.016
(b) 0.16
(c) 10
(d) 16

21. if $x = \frac{1}{4}$, what is the value of $\frac{2}{2-x}$?
(a) $\frac{3}{8}$
(b) $\frac{8}{9}$
(c) $1\frac{1}{7}$
(d) $3\frac{1}{2}$

22. if $n \times \frac{7}{12} = \frac{7}{12} \times \frac{3}{14}$, then $\frac{1}{n} =$
(a) $\frac{14}{3}$
(b) $\frac{12}{7}$
(c) $\frac{1}{6}$
(d) $\frac{3}{14}$

Question

\\frac{10}{13} = 0.\overline{769230}\

15. in the infinitely repeating decimal above, 7 is the first digit in the repeating pattern. what is the $391^{\text{st}}$ digit?
(a) 0
(b) 3
(c) 6
(d) 7

16. bryan completed a 100-meter race in 11.74 seconds. luis completed the same race in 11.69 seconds. what was the difference between their times, expressed as a fraction of a second?
(a) $\frac{1}{20}$
(b) $\frac{1}{10}$
(c) $\frac{3}{20}$
(d) $\frac{1}{5}$

\6.3\overline{4} - 4.1\overline{2}\

17. what is the value of the expression above, written as an improper fraction?
(a) $\frac{24}{11}$
(b) $\frac{224}{101}$
(c) $\frac{111}{50}$
(d) $\frac{20}{9}$

18. the decimal $0.0\overline{6}$ can be written as the fraction $\frac{x}{90}$. what is the value of $x$?
(a) 3
(b) 6
(c) 12
(d) 30

19. what decimal is equivalent to $-\frac{13}{11}$?
(a) $-1.18$
(b) $-1.118$
(c) $-1.\overline{18}$
(d) $-1.\overline{81}$

20. the decimal $0.4$ can be written as the fraction $\frac{x}{25}$. what is the value of $x$?
(a) 0.016
(b) 0.16
(c) 10
(d) 16

21. if $x = \frac{1}{4}$, what is the value of $\frac{2}{2-x}$?
(a) $\frac{3}{8}$
(b) $\frac{8}{9}$
(c) $1\frac{1}{7}$
(d) $3\frac{1}{2}$

22. if $n \times \frac{7}{12} = \frac{7}{12} \times \frac{3}{14}$, then $\frac{1}{n} =$
(a) $\frac{14}{3}$
(b) $\frac{12}{7}$
(c) $\frac{1}{6}$
(d) $\frac{3}{14}$

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Fractional Subtraction", "Simplifying Fractions", "Algebraic Substitution" ], "new_concepts": [ "Repeating Decimals", "Fraction Decimal Conversion", "Reciprocal Equations" ], "current_concepts": [ "Repeating Decimals", "Fractional Subtraction", "Simplifying Fractions", "Fraction Decimal Conversion", "Algebraic Substitution", "Reciprocal Equations" ] } ## Question 15: Find the 391st digit of the repeating decimal $$ \begin{aligned} &\frac{10}{13} = 0.\overline{769230} \quad (\text{period length } L = 6) \ &391 \pmod 6 = 1 \ &\text{The 1st digit in the pattern is } 7 \end{aligned} $$ ## Question 16: Calculate the difference in race times as a fraction $$ \begin{aligned} &\Delta t = 11.74 - 11.69 = 0.05\text{ seconds} \ &0.05 = \frac{5}{100} = \frac{1}{20} \end{aligned} $$ ## Question 17: Evaluate the difference of repeating decimals as an improper fraction $$ \begin{aligned} &6.\overline{34} - 4.\overline{12} = 2.\overline{22} \ &2.\overline{22} = 2 + \frac{22}{99} = 2 + \frac{2}{9} = \frac{20}{9} \end{aligned} $$ ## Question 18: Solve for x in the decimal-to-fraction equation $$ \begin{aligned} &0.06 = \frac{6}{100} = \frac{3}{50} \ &\frac{3}{50} = \frac{x}{50} \implies x = 3 \end{aligned} $$ ## Question 19: Convert the fraction to a decimal $$ \begin{aligned} &-\frac{13}{11} = -1.181818\dots = -1.\overline{18} \end{aligned} $$ ## Question 20: Solve for x in the decimal-to-fraction equation $$ \begin{aligned} &0.4 = \frac{4}{10} = \frac{2}{5} = \frac{10}{25} \ &\frac{x}{25} = \frac{10}{25} \implies x = 10 \end{aligned} $$ ## Question 21: Evaluate the algebraic expression $$ \begin{aligned} &x = \frac{1}{4} \ &\frac{2}{2 - x} = \frac{2}{2 - \frac{1}{4}} = \frac{2}{\frac{7}{4}} = \frac{8}{7} = 1\frac{1}{7} \end{aligned} $$ ## Question 22: Solve for the reciprocal of N $$ \begin{aligned} &N \times \frac{7}{12} = \frac{7}{12} \times \frac{3}{14} \implies N = \frac{3}{14} \ &\frac{1}{N} = \frac{14}{3} \end{aligned} $$ ## Question 15 (A) 0

(B) 3

(C) 6

(D) 7 ## Question 16 (A) $\frac{1}{20}$

(B) $\frac{1}{10}$

(C) $\frac{3}{20}$

(D) $\frac{1}{5}$ ## Question 17 (A) $\frac{24}{11}$

(B) $\frac{224}{101}$

(C) $\frac{111}{50}$

(D) $\frac{20}{9}$ ## Question 18 (A) 3

(B) 6

(C) 12

(D) 30 ## Question 19 (A) -1.18

(B) -1.118

(C) -1.\overline{18}

(D) -1.\overline{81} ## Question 20 (A) 0.016

(B) 0.16

(C) 10

(D) 16 ## Question 21 (A) $\frac{3}{4}$

(B) $\frac{8}{9}$

(C) $1\frac{1}{7}$

(D) $3\frac{1}{2}$ ## Question 22 (A) $\frac{14}{3}$

(B) $\frac{12}{7}$

(C) $\frac{1}{6}$

(D) $\frac{3}{14}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Fraction Decimal Conversion" ] }

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QUESTION IMAGE

Question

Explanation:

Question 15: Find the 391st digit of the repeating decimal

Question 16: Calculate the difference in race times as a fraction

Question 17: Evaluate the difference of repeating decimals as an improper fraction

Question 18: Solve for x in the decimal-to-fraction equation

Question 19: Convert the fraction to a decimal

Question 20: Solve for x in the decimal-to-fraction equation

Question 21: Evaluate the algebraic expression

Question 22: Solve for the reciprocal of N

Answer:

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22