The dimensions of Lincoln County on the map are $\displaystyle 4 \frac{1}{4} = \frac{17}{4}$ inches by $\displaystyle 3 \frac{3}{4} = \frac{15}{4}$ inches.
 

One half of an inch represents $\displaystyle N$ miles, so one whole inch represents twice this, or $\displaystyle 2N$ miles. Consequently,

$\displaystyle \frac{17}{4}$ inches represents $\displaystyle \frac{17}{4} \cdot 2N = \frac{17}{2} N$ miles, and

$\displaystyle \frac{15}{4}$ inches represents $\displaystyle \frac{15}{4} \cdot 2N = \frac{15}{2} N$ miles.

Therefore, in real distance, Lincoln County has dimensions $\displaystyle \frac{17}{2} N$ miles and $\displaystyle \frac{15}{2} N$ miles, and the area is the product of these two, or

$\displaystyle \frac{17}{2} N \cdot \frac{15}{2} N = \frac{255}{4}N^{2}$.

In order to make seven butter cakes, Floyd must use

$\displaystyle 4 \times 7 = 28$ eggs.

Floyd has three dozen, or 36, eggs on hand, so he has enough eggs.

Floyd must use

$\displaystyle 2 \frac{1}{2} \times 7 = \frac{5}{2} \times \frac{7}{1} = \frac{35}{2} = 17 \frac{1}{2}$ cups of flour.

Floyd has 16 cups on hand, so he does not have enough flour.

The formula for multiplication of fractions is:

$\displaystyle \frac{a}{b\ }\times \frac{c}{d} =\frac{ac}{bd}$

Where $\displaystyle b\neq 0$ and $\displaystyle d\neq 0.$ 

Convert each mixed number to an improper fraction; then multiply.

$\displaystyle 3\frac{1}{2} = \frac{(2\times 3)+1}{2} = \frac{7}{2}$

$\displaystyle 2\frac{2}{3} = \frac{(3\times 2) +2}{3} = \frac{8}{3}$

It doesn't matter if the denominators are not the same. Just multiply the numerators, then multiply the denominators and simplify.

$\displaystyle \frac{7}{2} \times \frac{8}{3} = \frac{7\times 8}{2\times 3} = \frac{56}{6}$

Convert the improper fraction back to a mixed number by dividing the numerator by the denominator.

$\displaystyle 56\div6 = 9\frac{2}{6}$

Simplify by dividing the numerator and denominator of the fraction part of the mixed number by the GCF or Greatest Common Factor which is 2.

$\displaystyle \frac{2\div 2}{6\div2 } = \frac{1}{3}$

$\displaystyle 9 \frac{2}{6}$ reduced to simplest form is $\displaystyle 9\frac{1}{3}.$

To simplify, divide the numerator and the denominator by the GCF or the greatest common factor, which is 15.

$\displaystyle -\frac{15\div 15}{30\div 15} =-\frac{1}{2}$

The formula for multiplication of fractions is:

$\displaystyle \frac{a}{b\ }\times \frac{c}{d} =\frac{ac}{bd}$

Where $\displaystyle b\neq 0$ and $\displaystyle d\neq 0.$ It doesn't matter if the denominators are not the same. Just multiply the numerators, then multiply the denominators and simplify.

$\displaystyle \frac{4}{9\ }\times \frac{3}{5} =\frac{4\times 3}{9\times 5}$

$\displaystyle \frac{4\times 3}{9\times 5} = \frac{12}{45}$

To simplify, divide the numerator and the denominator by the GCF or the greatest common factor, which is 3.

$\displaystyle \frac{12\div 3}{45\div 3} =\frac{4}{15}$

The formula for multiplication of fractions is:

$\displaystyle \frac{a}{b\ }\times \frac{c}{d} =\frac{ac}{bd}$

Where $\displaystyle b\neq 0$ and $\displaystyle d\neq 0.$ It doesn't matter if the denominators are not the same. Just multiply the numerators, then multiply the denominators and simplify.

Convert any mixed number (whole number and fraction) to an improper fraction:

$\displaystyle 6\frac{8}{9} = \frac{(9\times6)+8}{9} = \frac{62}{9}$

$\displaystyle \frac{62}{9 }\times \frac{3}{8} = \frac{62\times 3}{9\times 8}$

$\displaystyle \frac{186}{72} = 2\frac{42}{72}$

To simplify, divide the numerator and the denominator by the GCF, or the greatest common factor, which is 6.

$\displaystyle 2\frac{42}{72} = 2\frac{42\div 6}{72\div 6} =$

$\displaystyle 2\frac{7}{12}$

The formula for Area of a rectangle is $\displaystyle A = l\times w$, where $\displaystyle l = length$ and $\displaystyle w = width.$

To solve $\displaystyle 3\frac{5}{9} \times 2\frac{1}{4}$, first convert both mixed numbers to improper fractions.

$\displaystyle 3\frac{5}{9} = \frac{(9 \times 3)+5}{9} = \frac{32}{9}$

$\displaystyle 2\frac{1}{4} = \frac{(4\times 2) +1}{2} = \frac{9}{2}$

Then multiply the numerators and the denominators. Simplify.

$\displaystyle \frac{32}{9} \times \frac{9}{4} =\frac{32\times 9}{9\times 4}$

Because there is a 9 in the numerator and a 9 in the denominator the nines cancel each other out, and you are left with:

$\displaystyle \frac{32}{4} = 8$

Area is 2-dimensional: it has a length and a width. Area is measured in square units.

The area of the rectangle is $\displaystyle 8cm^{2}$. 

To solve this word problem, multiply the fractions.  First convert any mixed number to an improper fraction:

$\displaystyle 13\frac{1}{2} = \frac{(2\times13 )+1}{2} = \frac{27}{2}$

Next, multiply the numerators and the denominators of both improper fractions.

$\displaystyle \frac{27}{2} \times\frac{2}{3} = \frac{54}{6} = 9$

$\displaystyle 9$ gallons of gas was consumed on Renee's trip to the shore.

The formula for the Area of a triangle is:

$\displaystyle A= \frac{1}{2}(b\times h)$ where b represents the base and h represents height.

Convert any mixed numbers to improper fractions.

$\displaystyle 1\frac{1}{8} = \frac{(8\times 1)+1}{8} = \frac{9}{8}$

Plug in the values for the base and the height into the formula; multiply.

$\displaystyle A = \frac{1}{2} (\frac{2}{3} \times \frac{9}{8})$

$\displaystyle A = \frac{(1\times 2\times 9)}{(2\times 3\times 8}) = \frac{18}{48}$

Reduce to simplest form by dividing the numerator and the denominator by the Greatest Common Factor (GCF) of 18 and 48, which is 6.

$\displaystyle 18\div6 =3$

$\displaystyle 48\div6 = 8$

Because a triangle is two-dimensional, the answer must be represented in square units.

$\displaystyle A = \frac{3}{8} in^{2}$

The formula for the Volume of a rectangular prism is:$\displaystyle V = l\times w \times h$, where 

$\displaystyle l= length$,  $\displaystyle w = width$  and  $\displaystyle h = height.$

$\displaystyle V = 2\frac{1}{2} \times 1\frac{1}{3} \times \frac{3}{4}$

Convert mixed numbers (whole number and fraction) to improper fractions.

$\displaystyle 2\frac{1}{2}= \frac{(2 \times 2)+1}{2} = \frac{5}{2}$

$\displaystyle 1\frac{1}{3} = \frac{(3\times 1)+1}{3} = \frac{4}{3}$

$\displaystyle V = \frac{5}{2} \times \frac{4}{3}\times\frac{3}{4}$

$\displaystyle V = \frac{60}{24} cm^{3}$

$\displaystyle V = 2\frac{12}{24}cm^{3}$

Reduce to simplest form by dividing the numerator and the denominator of the fraction by the Greatest Common Factor of 12 and 24, which is 12.

A rectangular prism is a 3-dimensional shape, therefore the measurement must be expressed in cubic units.

$\displaystyle V = 2\frac{1}{2}cm^{3}$

The formula for multiplication of fractions is:

$\displaystyle \frac{a}{b\ }\times \frac{c}{d} =\frac{ac}{bd}$

Where $\displaystyle b\neq 0$ and $\displaystyle d\neq 0.$ It doesn't matter if the denominators are not the same. Just multiply the numerators, then multiply the denominators and simplify.

$\displaystyle -\frac{5}{6\ }\times \frac{3}{5} =\frac{-5\times 3}{6\times 5}$

$\displaystyle \frac{-5\times 3}{6\times 5} = -\frac{15}{30}$   A negative times a positive is a negative.

To simplify, divide the numerator and the denominator by the GCF or the greatest common factor, which is 15.

$\displaystyle -\frac{15\div 15}{30\div 15} =-\frac{1}{2}$