Common Core: 6th Grade Math : The Number System

Study concepts, example questions & explanations for Common Core: 6th Grade Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Interpret And Compute Quotients Of Fractions: Ccss.Math.Content.6.Ns.A.1

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{5}\ miles\) in length and occupies an area of \(\displaystyle \frac{7}{8}\ miles^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 6\tfrac{3}{8}\ miles\)

\(\displaystyle 3\tfrac{3}{8}\ miles\)

\(\displaystyle 4\tfrac{3}{8}\ miles\)

\(\displaystyle 4\tfrac{1}{8}\ miles\)

\(\displaystyle 4\tfrac{5}{8}\ miles\)

Correct answer:

\(\displaystyle 4\tfrac{3}{8}\ miles\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{7}{8}\ miles^2=\frac{1}{5}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{5}\ miles\)

\(\displaystyle \frac{\frac{7}{8}\ miles^2}{\frac{1}{5}\ miles}=\frac{\frac{1}{5}\ miles \times Width}{\frac{1}{5}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{5}\ miles\)

\(\displaystyle \frac{1}{5}\ miles\rightarrow \frac{5}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{5}{1} \times \frac{7}{8}\)

Multiply and solve.

\(\displaystyle Width=\frac{35}{8}\)

Reduce.

\(\displaystyle Width=4\tfrac{3}{8}\ miles\)

The width of the fracking site is \(\displaystyle 4\tfrac{3}{8}\ miles\)

Example Question #2 : Interpret And Compute Quotients Of Fractions: Ccss.Math.Content.6.Ns.A.1

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{4}\ miles\) in length and occupies an area of \(\displaystyle \frac{7}{9}\ miles^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 3\tfrac{5}{9}\ miles\)

\(\displaystyle 9\tfrac{5}{9}\ miles\)

\(\displaystyle 3\tfrac{1}{9}\ miles\)

\(\displaystyle 5\tfrac{1}{3}\ miles\)

\(\displaystyle 9\tfrac{1}{3}\ miles\)

Correct answer:

\(\displaystyle 3\tfrac{1}{9}\ miles\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{7}{9}\ miles^2=\frac{1}{4}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{4}\ miles\)

\(\displaystyle \frac{\frac{7}{9}\ miles^2}{\frac{1}{4}\ miles}=\frac{\frac{1}{4}\ miles \times Width}{\frac{1}{4}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{4}\ miles\)

\(\displaystyle \frac{1}{4}\ miles\rightarrow \frac{4}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{4}{1} \times \frac{7}{9}\)

Multiply and solve.

\(\displaystyle Width=\frac{28}{9}\)

Reduce.

\(\displaystyle Width=3\tfrac{1}{9}\ miles\)

The width of the fracking site is \(\displaystyle 3\tfrac{1}{9}\ miles\)

Example Question #3 : Interpret And Compute Quotients Of Fractions: Ccss.Math.Content.6.Ns.A.1

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{4}\ miles\) in length and occupies an area of \(\displaystyle \frac{7}{8}\ miles^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 3\tfrac{2}{3}\ miles\)

\(\displaystyle 2\tfrac{2}{3}\ miles\)

\(\displaystyle 3\tfrac{1}{3}\ miles\)

\(\displaystyle 3\tfrac{1}{2}\ miles\)

\(\displaystyle 2\tfrac{1}{3}\ miles\)

Correct answer:

\(\displaystyle 3\tfrac{1}{2}\ miles\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{7}{8}\ miles^2=\frac{1}{4}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{4}\ miles\)

\(\displaystyle \frac{\frac{7}{8}\ miles^2}{\frac{1}{4}\ miles}=\frac{\frac{1}{4}\ miles \times Width}{\frac{1}{4}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{4}\ miles\)

\(\displaystyle \frac{1}{4}\ miles\rightarrow \frac{4}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{4}{1} \times \frac{7}{8}\)

Multiply and solve.

\(\displaystyle Width=\frac{28}{8}\)

Reduce.

\(\displaystyle Width=3\tfrac{1}{2}\ miles\)

The width of the fracking site is \(\displaystyle 3\tfrac{1}{2}\ miles\)

Example Question #1 : The Number System

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{5}\ miles\) in length and occupies an area of \(\displaystyle \frac{35}{41}\ miles^2\). How wide is this particular site?

 

 
Possible Answers:

\(\displaystyle 2\tfrac{11}{41}\ miles\)

\(\displaystyle 5\tfrac{11}{41}\ miles\)

\(\displaystyle 4\tfrac{21}{41}\ miles\)

\(\displaystyle 4\tfrac{11}{41}\ miles\)

\(\displaystyle 11\tfrac{4}{41}\ miles\)

Correct answer:

\(\displaystyle 4\tfrac{11}{41}\ miles\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{35}{41}\ miles^2=\frac{1}{5}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{5}\ miles\)

\(\displaystyle \frac{\frac{35}{41}\ miles^2}{\frac{1}{5}\ miles}=\frac{\frac{1}{5}\ miles \times Width}{\frac{1}{5}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{5}\ miles\)

\(\displaystyle \frac{1}{5}\ miles\rightarrow \frac{5}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{5}{1} \times \frac{35}{41}\)

Multiply and solve.

\(\displaystyle Width=\frac{175}{41}\)

Reduce.

\(\displaystyle Width=4\tfrac{11}{41}\ miles\)

The width of the fracking site is \(\displaystyle 4\tfrac{11}{41}\ miles\)

Example Question #173 : How To Divide Fractions

Hydraulic fracturing is a process used by gas companies to rupture and collect pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{3}{4}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{1}{2}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle \frac{6}{7}\textup{ miles}\)

\(\displaystyle \frac{2}{3}\textup{ miles}\)

\(\displaystyle \frac{2}{5}\textup{ miles}\)

\(\displaystyle \frac{3}{5}\textup{ miles}\)

\(\displaystyle \frac{1}{3}\textup{ miles}\)

Correct answer:

\(\displaystyle \frac{2}{3}\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{1}{2}\ miles^2=\frac{3}{4}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{3}{4}\ miles\)

\(\displaystyle \frac{\frac{1}{2}\ miles^2}{\frac{3}{4}\ miles}=\frac{\frac{3}{4}\ miles \times Width}{\frac{3}{4}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{3}{4}\ miles\)

\(\displaystyle \frac{3}{4}\ miles\rightarrow \frac{4}{3}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{4}{3} \times \frac{1}{2}\)

Multiply and solve.

\(\displaystyle Width=\frac{4}{6}\)

Reduce.

\(\displaystyle Width=\frac{2}{3}\ miles\)

The width of the fracking site is \(\displaystyle \frac{2}{3}\ miles\)

Example Question #1 : Interpret And Compute Quotients Of Fractions: Ccss.Math.Content.6.Ns.A.1

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{7}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{3}{5}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 4\tfrac{3}{5}\textup{ miles}\)

\(\displaystyle 4\textup{ miles}\)

\(\displaystyle 4\tfrac{1}{5}\textup{ miles}\)

\(\displaystyle 4\tfrac{2}{5}\textup{ miles}\)

\(\displaystyle 4\tfrac{4}{5}\textup{ miles}\)

Correct answer:

\(\displaystyle 4\tfrac{1}{5}\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{3}{5}\ miles^2=\frac{1}{7}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{7}\ miles\)

\(\displaystyle \frac{\frac{3}{5}\ miles^2}{\frac{1}{7}\ miles}=\frac{\frac{1}{7}\ miles \times Width}{\frac{1}{7}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{7}\ miles\)

\(\displaystyle \frac{1}{7}\ miles\rightarrow \frac{7}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{7}{1} \times \frac{3}{5}\)

Multiply and solve.

\(\displaystyle Width=\frac{21}{5}\)

Reduce.

\(\displaystyle Width=4\tfrac{1}{5}\ miles\)

The width of the fracking site is \(\displaystyle 4\tfrac{1}{5}\ miles\)

Example Question #91 : Grade 6

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{2}{7}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{2}{5}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle \frac{3}{5}\textup{ miles}\)

\(\displaystyle 1\tfrac{2}{5}\textup{ miles}\)

\(\displaystyle 1\tfrac{3}{5}\textup{ miles}\)

\(\displaystyle 1\tfrac{5}{7}\textup{ miles}\)

\(\displaystyle 2\tfrac{3}{5}\textup{ miles}\)

Correct answer:

\(\displaystyle 1\tfrac{2}{5}\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{2}{5}\ miles^2=\frac{2}{7}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{2}{7}\ miles\)

\(\displaystyle \frac{\frac{2}{5}\ miles^2}{\frac{2}{7}\ miles}=\frac{\frac{2}{7}\ miles \times Width}{\frac{2}{7}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{2}{7}\ miles\)

\(\displaystyle \frac{2}{7}\ miles\rightarrow \frac{7}{2}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{7}{2} \times \frac{2}{5}\)

Multiply and solve.

\(\displaystyle Width=\frac{14}{10}\)

Reduce.

\(\displaystyle Width=1\tfrac{2}{5}\ miles\)

The width of the fracking site is \(\displaystyle 1\tfrac{2}{5}\ miles\)

Example Question #92 : Grade 6

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{5}{7}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{3}{5}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle \frac{31}{35}\textup{ miles}\)

\(\displaystyle \frac{21}{27}\textup{ miles}\)

\(\displaystyle \frac{24}{25}\textup{ miles}\)

\(\displaystyle \frac{21}{25}\textup{ miles}\)

\(\displaystyle \frac{11}{15}\textup{ miles}\)

Correct answer:

\(\displaystyle \frac{21}{25}\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{3}{5}\ miles^2=\frac{5}{7}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{5}{7}\ miles\)

\(\displaystyle \frac{\frac{3}{5}\ miles^2}{\frac{5}{7}\ miles}=\frac{\frac{5}{7}\ miles \times Width}{\frac{5}{7}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{5}{7}\ miles\)

\(\displaystyle \frac{5}{7}\ miles\rightarrow \frac{7}{5}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{7}{5} \times \frac{3}{5}\)

Multiply and solve.

\(\displaystyle Width=\frac{21}{25}\)

The width of the fracking site is \(\displaystyle \frac{21}{25}\ miles\)

Example Question #2 : Interpret And Compute Quotients Of Fractions: Ccss.Math.Content.6.Ns.A.1

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{6}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{5}{8}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 5\tfrac{3}{4}\textup{ miles}\)

\(\displaystyle 4\tfrac{1}{4}\textup{ miles}\)

\(\displaystyle 3\tfrac{5}{7}\textup{ miles}\)

\(\displaystyle 3\tfrac{1}{3}\textup{ miles}\)

\(\displaystyle 3\tfrac{3}{4}\textup{ miles}\)

Correct answer:

\(\displaystyle 3\tfrac{3}{4}\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{5}{8}\ miles^2=\frac{1}{6}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{6}\ miles\)

\(\displaystyle \frac{\frac{5}{8}\ miles^2}{\frac{1}{6}\ miles}=\frac{\frac{1}{6}\ miles \times Width}{\frac{1}{6}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{6}\ miles\)

\(\displaystyle \frac{1}{6}\ miles\rightarrow \frac{6}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{6}{1} \times \frac{5}{8}\)

Multiply and solve.

\(\displaystyle Width=\frac{30}{8}\)

Reduce.

\(\displaystyle Width=3\tfrac{3}{4}\ miles\)

The width of the fracking site is \(\displaystyle 3\tfrac{3}{4}\ miles\)

Example Question #94 : Grade 6

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is \(\displaystyle \frac{1}{6}\textup{ miles}\) in length and occupies an area of \(\displaystyle \frac{5}{6}\textup{ miles}^2\). How wide is this particular site?

Possible Answers:

\(\displaystyle 4\textup{ miles}\)

\(\displaystyle 5\tfrac{2}{5}\textup{ miles}\)

\(\displaystyle 5\textup{ miles}\)

\(\displaystyle 5\tfrac{3}{5}\textup{ miles}\)

\(\displaystyle 5\tfrac{1}{5}\textup{ miles}\)

Correct answer:

\(\displaystyle 5\textup{ miles}\)

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

\(\displaystyle \text{Area}=\text{Length} \times \text{Width}\)

Substitute in the given values in the equation and solve for \(\displaystyle \text{Width}\).

\(\displaystyle \frac{5}{6}\ miles^2=\frac{1}{6}\ miles\times Width\)

Divide both sides by \(\displaystyle \frac{1}{6}\ miles\)

\(\displaystyle \frac{\frac{5}{6}\ miles^2}{\frac{1}{6}\ miles}=\frac{\frac{1}{6}\ miles \times Width}{\frac{1}{6}\ miles}\)

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of \(\displaystyle \frac{1}{6}\ miles\)

\(\displaystyle \frac{1}{6}\ miles\rightarrow \frac{6}{1}\ miles\)

Simplify and rewrite.

\(\displaystyle Width=\frac{6}{1} \times \frac{5}{6}\)

Multiply and solve.

\(\displaystyle Width=\frac{30}{6}\)

Reduce.

\(\displaystyle Width=5\ miles\)

The width of the fracking site is \(\displaystyle 5\ miles\)

Learning Tools by Varsity Tutors