Problem 1

Write 1.23456 as an addition of a series of fractional powers of 10 and then as a series of simplified fractions.

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Problem 2

Find the sum of

\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}

Problem 3

Try to find the product of

\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}

without performing the actual multiplication.

Problem 4

Solve the equation

\frac{1+2+3+4}{a}+\frac{1+2+3+4+5}{a}=5

Problem 5

Solve the equation

\frac{41}{7}-\frac{a}{14}=4

Problem 6

Find a Rational number in between

a. \frac{1}{4} and \frac{1}{3}

b. \frac{5}{8} and \frac{3}{4}

c. -\frac{2}{5} and -\frac{3}{5}

Problem 7

Divide 21 into \frac{1}{5}'s and give each fractional value in both improper and mixed fractional form.

Problem 8

Divide 24 into \frac{1}{10}'s and give each fractional value in both improper and mixed fractional form.

Problem 9

Divide the number line interval of \frac{1}{6} to \frac{5}{6} into 4 equal spaces and give the values of the endpoints of each space.

Problem 10

Divide the number line interval of \frac{6}{5} to \frac{8}{5} into 10 equal spaces and give the values of the endpoints of each space.

Problem 11

Name at least 4 Rational numbers that are between

a. \frac{1}{2} and 1\frac{1}{2}

b. \frac{1}{4} and \frac{1}{2}

c. 0 and \frac{1}{10}

Problem 12

Solve:

1-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}

Problem 13

How many \frac{1}{5}'s are in

a. \frac{16}{5}

b. 41\frac{4}{5}

c. 2

d. \frac{3}{10}

Problem 14

What is the fractional distance between these sets of points [a,b] on the number line?

Interval 1: a=\frac{14}{3} and b=\frac{21}{3}

Interval 2: a=4\frac{3}{7} and b=8\frac{2}{7}

Interval 3: a=\frac{7}{3} and b=4\frac{1}{3}

Interval 4: a=2\frac{1}{3} and b=5\frac{1}{6}

Problem 15

What is the midpoint of each of these intervals on the number line?

a. \frac{1}{10} and \frac{7}{10}

b. \frac{17}{15} and \frac{20}{15}

c. \frac{3}{5} and \frac{2}{3}

Problem 16

How many Rational numbers are there between -1\leq x\leq1?

Problem 17

What happens to the denominator and overall value of the Rational number \frac{1}{x} when

a. x is less than 1 but greater than 0 and x is approaching 0

(x: 0<x<1)

b. x is greater than -1 but less than 0 and it is approaching 0 (approaching 0 from the negative side)

(x:-1<x<0)

c. x is less than -1 and approaches -1000

(x:x<-1 and x\to -1000)

Problem 1 Solution

Write 1.23456 as an addition of a series of fractional powers of 10 and then again as a series of simplified fractions.

Powers of 10: 1.23456=\frac{1}{1}+\frac{2}{10}+\frac{3}{100}+\frac{4}{1000}+\frac{5}{10000}+\frac{6}{100000}

Simplified Fractions: 1.23456=\frac{1}{1}+\frac{1}{5}+\frac{3}{100}+\frac{1}{250}+\frac{1}{2000}+\frac{3}{50000}

Problem 2 Solution

Find the sum of

\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}

\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{30}{60}+\frac{20}{60}+\frac{15}{60}+\frac{12}{60}+\frac{10}{60}=\frac{87}{60}=\frac{29}{20}

Problem 3 Solution

Try to find the product of

\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}

without performing the actual multiplication.

\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}=\frac{1}{6}

The denominator of each fraction cancels with the numerator of the fraction following it.

The denominator 2 in \frac{1}{2} cancels the numerator 2 in \frac{2}{3} etc.

The only 2 numbers that don't cancel are the numerator 1 in \frac{1}{2} and the denominator 6 in \frac{5}{6} leaving \frac{1}{6}

We can also rearrange the numerators to make 4 of the fractions equivalent to 1

\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}= \frac{2}{2}\times\frac{3}{3}\times\frac{4}{4}\times\frac{5}{5}\times\frac{1}{6}=1\times1\times1\times1\times\frac{1}{6}=\frac{1}{6}

Problem 4 Solution

Solve the equation

\frac{1+2+3+4}{a}+\frac{1+2+3+4+5}{a}=5

\frac{1+2+3+4}{a}+\frac{1+2+3+4+5}{a}=\frac{10}{a}+\frac{15}{a}=\frac{25}{a}=5\implies a=5

Problem 5 Solution

Solve the equation

\frac{41}{7}-\frac{a}{14}=4

\frac{41}{7}-\frac{a}{14}=4\implies\frac{41}{7}-4=\frac{a}{14}

\frac{41}{7}-4=\frac{41}{7}-\frac{28}{7}=\frac{13}{7}\implies

\frac{13}{7}=\frac{a}{14}\implies a=(14)\frac{13}{7}=(2)(13)=26

Problem 6 Solution

Find a Rational number in between

a. \frac{1}{4} and \frac{1}{3}

\frac{1}{4}=\frac{3}{12} and \frac{1}{3}=\frac{4}{12}

There is no fraction with a denominator of 12 between \frac{3}{12} and \frac{4}{12}.

Converting denominators gives \frac{3}{12}=\frac{6}{24} and \frac{4}{12}=\frac{8}{24}

There is a fraction in between \frac{6}{24} and \frac{8}{24},

namely \frac{7}{24}

\frac{1}{4}=\frac{6}{24}<\frac{7}{24}<\frac{8}{24}=\frac{1}{3}

This is just one of infinitely many fractions between \frac{1}{4} and \frac{1}{3}.

b. \frac{5}{8} and \frac{3}{4}

\frac{5}{8} and \frac{3}{4}=\frac{6}{8}

\frac{5}{8}=\frac{10}{16} and \frac{6}{8}=\frac{12}{16}

There is a fraction between \frac{10}{16} and \frac{12}{16},

namely \frac{11}{16}.

c. -\frac{3}{5} and -\frac{2}{5}

-\frac{3}{5}=-\frac{6}{10} and -\frac{2}{5}=-\frac{4}{10}

-\frac{5}{10}=-\frac{1}{2}

-\frac{3}{5}<-\frac{1}{2}<-\frac{2}{5}

Problem 7 Solution

Divide 21 into \frac{1}{5}'s and give each fractional value in improper and mixed fraction form.

Interval 0\to 21 into \frac{1}{5}'s

1. \frac{21}{5}=4\frac{1}{5}

 2. \frac{42}{5}=8\frac{2}{5}

  3. \frac{63}{5}=12\frac{3}{5}

  4. \frac{84}{5}=16\frac{4}{5}

                                                                                                    5. \frac{105}{5}=21

Problem 8 Solution

Divide 24 into \frac{1}{10}'s and give each fractional value in both improper and mixed form.

Interval 0\to 24 into \frac{1}{10}'s

1. \frac{24}{10}=\frac{12}{5}=2\frac{2}{5}

 2. \frac{48}{10}=\frac{24}{5}=4\frac{4}{5}

3. \frac{72}{10}=\frac{36}{5}=7\frac{1}{5}

 4. \frac{96}{10}=\frac{48}{5}=9\frac{3}{5}

                                                                                       5. \frac{120}{10}=\frac{60}{5}=12

     6. \frac{144}{10}=\frac{72}{5}=14\frac{2}{5}

     7. \frac{168}{10}=\frac{84}{5}=16\frac{4}{5}

    8. \frac{192}{10}=\frac{96}{5}=19\frac{1}{5}

      9. \frac{216}{10}=\frac{108}{5}=21\frac{3}{5}

                                                                                    10. \frac{240}{10}=\frac{120}{5}=24

Problem 9 Solution

Divide the number line interval of \frac{1}{6} to \frac{5}{6} into \frac{1}{4}'s and give each value.

Number Line Distance: \frac{5}{6}-\frac{1}{6}=\frac{4}{6}

Spaces: \frac{\frac{4}{6}}{4}=\frac{1}{6}

Interval 1: \frac{1}{6}\to\frac{2}{6}\implies\frac{1}{6}\to\frac{1}{3}

Interval 2: \frac{1}{3}\to\frac{3}{6}\implies\frac{1}{3}\to\frac{1}{2}

Interval 3: \frac{1}{2}\to\frac{4}{6}\implies\frac{1}{2}\to\frac{2}{3}

                                                              Interval 4: \frac{2}{3}\to\frac{5}{6}

Problem 10 Solution

Divide the number line interval of \frac{6}{5} to \frac{8}{5} into \frac{1}{10}'s and give each value.

Number Line Distance: \frac{8}{5}-\frac{6}{5}=\frac{2}{5}

Spaces: \frac{\frac{2}{5}}{10}=\frac{1}{25}

Number Line Start and Finish: \frac{6}{5}=\frac{30}{25} and \frac{8}{5}=\frac{40}{25}

Interval 1: \frac{30}{25}\to\frac{31}{25}\implies\frac{6}{5}\to\frac{31}{25}

                                                        Interval 2: \frac{31}{25}\to\frac{32}{25}

                                                        Interval 3: \frac{32}{25}\to\frac{33}{25}

                                                        Interval 4: \frac{33}{25}\to\frac{34}{25}

 Interval 5: \frac{34}{25}\to\frac{35}{25}\implies\frac{34}{25}\to\frac{7}{5}

                                                        Interval 6: \frac{7}{5}\to\frac{36}{25}

Interval 7: \frac{36}{25}\to\frac{37}{25}

Interval 8: \frac{37}{25}\to\frac{38}{25}

Interval 9: \frac{38}{25}\to\frac{39}{25}

Interval 10: \frac{39}{25}\to\frac{40}{25}\implies\frac{39}{25}\to\frac{8}{5}

Problem 11 Solution

Name at least 4 Rational numbers that are between

a. \frac{1}{2} and 1\frac{1}{2}

Here are quite a few Rational numbers between \frac{1}{2} and 1\frac{1}{2}

\frac{3}{5}, \frac{3}{4}, \frac{4}{7}, \frac{5}{8}, \frac{2}{3},\frac{7}{8}, \frac{9}{10}, \frac{11}{10}, \frac{5}{4}, \frac{4}{3}, \frac{11}{8}

b. \frac{1}{4} and \frac{1}{2}

\frac{1}{4}\to\frac{2}{4}

\frac{2}{8}\to\frac{4}{8}

\frac{2}{8}<\frac{3}{8}<\frac{4}{8}

\frac{4}{16}\to\frac{8}{16}

\frac{4}{16}<\frac{5}{16}<\frac{7}{16}<\frac{8}{16}

\frac{8}{32}<\frac{9}{32}<\frac{11}{32}<\frac{13}{32}<\frac{15}{32}<\frac{16}{32}

c. 0 and \frac{1}{10}

0\to\frac{1}{10}

0  ...  <\frac{1}{17}<\frac{1}{16}<\frac{1}{15}<\frac{1}{14}<\frac{1}{13}<\frac{1}{12}<\frac{1}{11}<\frac{1}{10}

Problem 12 Solution

Solve:

1-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}

1-\frac{1}{1-\frac{1}{1-\frac{1}{2}}}

1-\frac{1}{1-\frac{1}{\frac{1}{2}}}

=1-\frac{1}{1-2}=1-\frac{1}{-1}=1+1=2

Problem 13 Solution

How many \frac{1}{5}'s are in

a. \frac{16}{5}

\frac{\frac{16}{5}}{\frac{1}{5}}=\frac{16}{5}\times\frac{5}{1}=16

b. 41\frac{4}{5}

41\frac{4}{5}=\frac{209}{5}

\frac{\frac{209}{5}}{\frac{1}{5}}=\frac{209}{5}\times\frac{5}{1}=209

c. 2

\frac{\frac{2}{1}}{\frac{1}{5}}=\frac{2}{1}\times\frac{5}{1}=10

d. \frac{3}{10}

\frac{\frac{3}{10}}{\frac{1}{5}}=\frac{3}{10}\times\frac{5}{1}=\frac{15}{10}=\frac{3}{2}=1\frac{1}{2}

Problem 14 Solution

What is the fractional distance d between these sets of points [a,b] on the number line?

Interval 1: a=\frac{14}{3} and b=\frac{21}{3}

d=b-a

[a,b]=[\frac{7}{3},\frac{14}{3}]

d=\frac{14}{3}-\frac{7}{3}=\frac{7}{3}=2\frac{1}{3}

Interval 2: a=4\frac{3}{7} and b=8\frac{2}{7}

4\frac{3}{7}=\frac{31}{7} and 8\frac{2}{7}=\frac{58}{7}

d=b-a

[a,b]=[\frac{31}{7},\frac{58}{7}]

d=\frac{58}{7}-\frac{31}{7}=\frac{27}{7}=3\frac{1}{7}

Interval 3: a=-\frac{4}{3} and b=4\frac{1}{3}

4\frac{1}{3}=\frac{13}{3}

d=b-a

[a,b]=[-\frac{4}{3},\frac{13}{3}]

d=\frac{13}{3}-(-\frac{4}{3})=\frac{13}{3}+\frac{4}{3}=\frac{17}{3}=5\frac{2}{3}

Interval 4: =2\frac{1}{3} and b=5\frac{1}{6}

2\frac{1}{3}=2\frac{2}{6}=\frac{14}{6} and 5\frac{1}{6}=\frac{31}{6}

d=b-a

[a,b]=[\frac{14}{6},\frac{31}{6}]

d=\frac{31}{6}-\frac{14}{6}=\frac{17}{6}=2\frac{5}{6}

Problem 15 Solution

What is the midpoint m of each of these intervals on the number line?

a. [a,b]=[\frac{1}{10},\frac{7}{10}]

m=\frac{a+b}{2}

m=\frac{\frac{7}{10}+\frac{1}{10}}{2}=\frac{\frac{8}{10}}{2}=\frac{8}{20}=\frac{4}{10}=\frac{2}{5}

b. [a,b]=[\frac{17}{15},\frac{20}{15}]

m=\frac{a+b}{2}

a=\frac{17}{15}=\frac{34}{30} and b=\frac{20}{15}=\frac{40}{30}

m=\frac{\frac{34}{30}+\frac{40}{30}}{2}=\frac{74}{60}=\frac{37}{30}

c. [a,b]=[\frac{3}{5},\frac{2}{3}]

m=\frac{a+b}{2}

a=\frac{3}{5}=\frac{18}{30} and b=\frac{2}{3}=\frac{20}{30}

m=\frac{\frac{18}{30}+\frac{20}{30}}{2}=\frac{38}{60}=\frac{19}{30}

Problem 16 Solution

How many Rational numbers x are there between -1\leq x\leq1?

There are an infinite amount of Rational numbers in between any 2 other Rational numbers.

Problem 17 Solution

What happens to the denominator and overall value of the Rational number \frac{1}{x} when

a. x is less than 1 but greater than 0 and x is approaching 0

(x: 0<x<1) with x\to0 or x\to0^{+}

The Rational number \frac{1}{x} gets bigger and bigger as x\to0.

x=\frac{1}{2} and (0<\frac{1}{2}<1) gives \frac{1}{x}=\frac{1}{\frac{1}{2}}=2

x=\frac{1}{4} and (0<\frac{1}{4}<1) gives \frac{1}{x}=\frac{1}{\frac{1}{4}}=4

 x=\frac{1}{10} and (0<\frac{1}{10}<1) gives \frac{1}{x}=\frac{1}{\frac{1}{10}}=10

x=\frac{1}{100} and (0<\frac{1}{100}<1) gives \frac{1}{x}=\frac{1}{\frac{1}{100}}=100

b. x is greater than -1 but less than 0 and x is approaching 0 (approaches 0 from the negative side)

(x:-1<x<0) with x\to0 or x\to0^{-}

The Rational number \frac{1}{x} gets bigger and bigger negatively as x\to0.

x=-\frac{1}{2} and (-1<-\frac{1}{2}<0) gives \frac{1}{x}=\frac{1}{-\frac{1}{2}}=-2

x=-\frac{1}{4} and (-1<-\frac{1}{4}<0) gives \frac{1}{x}=\frac{1}{-\frac{1}{4}}=-4

 x=-\frac{1}{10} and (-1<-\frac{1}{10}<0) gives \frac{1}{x}=\frac{1}{-\frac{1}{10}}=-10

x=-\frac{1}{100} and (-1<-\frac{1}{100}<0) gives \frac{1}{x}=\frac{1}{-\frac{1}{100}}=-100

c. x is less than -1 and approaches -1000

(x:x<-1) and x\to -1000

The Rational number \frac{1}{x} gets smaller and smaller negatively as x\to-1000.

x=-2 and (-2<-1) gives \frac{1}{x}=\frac{1}{-2}=-\frac{1}{2}

x=-4 and (-4<-1) gives \frac{1}{x}=\frac{1}{-4}=-\frac{1}{4}

 x=-10 and (-10<-1) gives \frac{1}{x}=\frac{1}{-10}=-\frac{1}{10}

x=-100 and (-100<-1) gives \frac{1}{x}=\frac{1}{-100}=-\frac{1}{100}

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.