How to multiply and divide fractions - HSPT Math

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Question

A tree feet high casts a shadow 20 feet long at the same time a tree 30 feet high casts a shadow how many feet long? Give your answer in terms of .

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Answer

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 :

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

Set the cross-products equal to each other and solve for

$M \cdot N = 20 \cdot 30$

$M \cdot N = 600$

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

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

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Question

Solve:

$\frac{2}{5}\times\frac{7}{8}=$

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Answer

$\frac{2}{5}\times\frac{7}{8}=\frac{2\times7}{5\times8}=\frac{14}{40}=\frac{7}{20}$

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Question

Solve:

$\frac{3}{4}\div \frac{1}{3}$

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Answer

$\frac{3}{4}\div\frac{1}{3}=\frac{3}{4}\times\frac{3}{1}=\frac{9}{4}$

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Question

Solve:

$\frac{3}{7}\div \frac{1}{5}$

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Answer

$\frac{3}{7}\div \frac{1}{5}=\frac{3}{7}\times \frac{5}{1}=\frac{15}{7}$

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Question

Solve:

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

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Answer

$\frac{2}{5}\times \frac{3}{4}=\frac{2\times3}{5\times4}=\frac{6}{20}=\frac{3}{10}$

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Question

The length and width of a given make of car are 4.5 and 1.8 meters, respectively.

A scale model of this make of car will be feet long. In terms of , how wide will this scale model be?

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Answer

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:

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

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

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

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Question

Edna needs $1\frac{1}{2}$ cups of sugar to make 24 of her world-famous Oatmeal Chocolate Dynamite Cookies.

She has cups of sugar on hand. If the other ingredients are sufficiently plentiful, then in terms of , how many cookies can she make? (Note: the number of cookies need not be a multiple of 24.)

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Answer

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

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

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

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

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

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Question

How many seconds are in $\frac{1}{15}$ of one minute?

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Answer

To find $\frac{1}{15}$ of one minute in terms of seconds, you need to first convert one minute to its value in seconds, which is 60 seconds.

Now, you just need to find the value for $\frac{1}{15}$ of 60, which you determine by multiplying the two values:

$\frac{1}{15} \times 60 = \frac{60}{15}$

You can simplify the above fraction by division, which will you give you your correct answer in seconds:

$\frac{60}{15} = 4$

Your answer is 4 seconds.

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Question

On average, 1 in 9 peanuts that are harvested will only have one pod instead of two pods. Of those one-pod peanuts, three quarters of them will be smaller than the average peanut.

If 900 peanuts are harvested, how many will be one-pod peanuts that are smaller than average in size?

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Answer

If there are 900 peanuts and 1 in 9 will be one-podded, that means that 100 will be one-podded.

$900\cdot \frac{1}{9}=100$

Three quarters of these 100 one-podded peanuts will be of smaller than average size. Three quarters of of 100 is 75. Therefore, 75 is the correct answer.

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Question

A small bakery sells two flavors of muffins: chocolate and banana.

At the end of the day, there are 13 muffins left.

There are chocolate muffins and banana muffins.

If

$x=\frac{7}{2}\cdot \frac{4}{7}-1$

how many banana muffins are left?

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Answer

The first step is to solve for x.

$x=\frac{7}{2}\cdot \frac{4}{7}-1$

If there are 1x chocolate muffins and 2y banana muffins, for a total of 13 muffins then:

Since the number of banana muffins remaining is 2y, there are 12 banana muffins left.

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Question

One Australian dollar is equal to about 86 cents. To the nearest whole, for how many Australian dollars can a tourist to Australia expect to exchange $600 in American money?

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Answer

One Australian dollar is equivalent to 86 cents, or $0.86, in American money, so divide $600 dollars by this conversion factor. $600 is equivalent to

$600 \div 0.86 \approx 698$ Australian dollars.

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Question

$\frac{4}{9}$ of a class of 36 students are boys. If 2 girls and 4 boys were to drop the class, what percentage of the class would be girls?

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Answer

First determine how many boys and girls are currently in the class.

$\frac{4}{9}\cdot 36=16$ boys in the class, which means that there are 20 girls.

When the 6 students drop the new class size will be 30, which will be made up of girls and boys.

$\frac{18}{30}=60%$

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Question

Multiply: $\frac{6}{7}\times \frac{1}{7}$

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Answer

Since we are multiply fractions, multiply the numerator with numerator, and denominator with denominator. Simplify if possible.

$\frac{6}{7}\times \frac{1}{7}= \frac{6}{49}$

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Question

Divide: $\frac{(\frac{2}{3})}{6}$

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Answer

Rewrite the expression using a division sign.

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

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

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

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Question

On a map, one half of an inch represents miles of real distance.

On the map, Lincoln County, which is perfectly rectangular, is four and one-fourth inches wide by three and three-fourths inches high. In terms of , give the area of Lincoln County in square miles.

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Answer

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

One half of an inch represents miles, so one whole inch represents twice this, or miles. Consequently,

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

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

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

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

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Question

One of Floyd's World Famous Ooey Gooey Butter Cakes requires two and one-half cups of flour and four eggs, among other ingredients.

Floyd wants to make seven of these cakes for a wedding. He has sixteen cups of flour on hand as well as three dozen eggs. Assuming he has enough of all other ingredients, can Floyd make the seven cakes he wants to make?

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Answer

In order to make seven butter cakes, Floyd must use

$4 \times 7 = 28$ eggs.

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

Floyd must use

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

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

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Question

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

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Answer

The formula for multiplication of fractions is:

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

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

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

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

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

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

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Question

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

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Answer

The formula for multiplication of fractions is:

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

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

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

$\frac{-5\times 3}{6\times 5} = -\frac{15}{30}$ 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.

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

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Question

$3\frac{1}{2 }\times 2\frac{2}{3}$

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Answer

The formula for multiplication of fractions is:

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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Question

$6 \frac{8}{9}\times\frac{3}{8}$

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Answer

The formula for multiplication of fractions is:

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

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

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

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

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

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

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

$2\frac{7}{12}$

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