Rewrite the expression using a division sign.

$\displaystyle \frac{2}{3}\div 6$

Convert the sign to a multiplication sign, and take the inverse of the second term.

$\displaystyle \frac{2}{3}\times \frac{1}{6} = \frac{1}{9}$

The ratio of the height of an object to the length of the shadow it casts at a point in time is the same for all objects, so, if we let the length of the shadow cast by the second tree be $\displaystyle N$:

$\displaystyle \frac{30}{N} = \frac{M}{20}$

Set the cross-products equal to each other and solve for $\displaystyle N$

$\displaystyle M \cdot N = 20 \cdot 30$

$\displaystyle M \cdot N = 600$

$\displaystyle M \cdot N\div M = 600 \div M$

$\displaystyle N = \frac{600}{M}$, the correct choice.

The ratio of the width of the scale model to its length is the same as the comparable ratio for the actual car, so we can set up this equation:

$\displaystyle \frac{W}{L} = \frac{1.8}{4.5} = \frac{1.8 \cdot 10 }{4.5\cdot 10 }= \frac{18 }{45}= \frac{18 \div 9}{45 \div 9}=\frac{2}{5}$

$\displaystyle \frac{W}{L} \cdot L =\frac{2}{5}\cdot L$

$\displaystyle W=\frac{2}{5} L$

An easier way to look at the number of cookies Edna can make with her sugar is as follows:

$\displaystyle 1\frac{1}{2}$ cups of sugar go into making 24 cookies, so

$\displaystyle 1\frac{1}{2} \times 2 = 3$ cups of sugar go into making $\displaystyle 24 \times 2 = 48$ cookies.

1 cup of sugar, therefore, goes into making $\displaystyle 48 \div 3 = 16$ cookies.

$\displaystyle N$ cups go into making $\displaystyle 16 \cdot N$, or $\displaystyle 16 N$, cookies. This is the correct response.

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 $\displaystyle 8cm^{2}$.