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HSPT Math : Fractions

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Example Questions

Example Question #5 : How To Multiply And Divide Fractions

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 .

Possible Answers:

$\frac{30M}{20}$

$\frac{M}{600}$

$\frac{600}{M}$

$\frac{20M}{30}$

Correct answer:

$\frac{600}{M}$

Explanation:

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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Example Question #43 : Arithmetic

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?

Possible Answers:

$\frac{10}{81}L$

$\frac{81}{10}L$

$\frac{2}{5} L$

$\frac{5}{2} L$

Correct answer:

$\frac{2}{5} L$

Explanation:

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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Example Question #6 : How To Multiply And Divide Fractions

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.)

Possible Answers:

$\frac{36}{N}$

36N

$\frac{16}{N}$

16N

Correct answer:

16N

Explanation:

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 16 N , cookies. This is the correct response.

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Example Question #43 : Arithmetic

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.

Possible Answers:

$\frac{255}{4}N^{2}$

$255N^{2}$

$\frac{255}{16}N^{2}$

$\frac{255}{2}N^{2}$

Correct answer:

$\frac{255}{4}N^{2}$

Explanation:

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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Example Question #42 : Problem Solving

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?

Possible Answers:

Yes.

No, because he does not have enough flour.

No, because he does not have enough eggs.

No, because he does not have enough flour or enough eggs.

Correct answer:

No, because he does not have enough flour.

Explanation:

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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Example Question #45 : Problem Solving

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

Possible Answers:

$9\frac{2}{3}$

$9\frac{1}{3}$

$6\frac{1}{3}$

$6\frac{2}{3}$

Correct answer:

$9\frac{1}{3}$

Explanation:

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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Example Question #41 : Arithmetic

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

Possible Answers:

$\frac{4}{15}$

$\frac{7}{15}$

$\frac{2}{3}$

$\frac{1}{4}$

Correct answer:

$\frac{4}{15}$

Explanation:

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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Example Question #45 : Problem Solving

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

Possible Answers:

$6\frac{3}{4}$

$6\frac{1}{3}$

$2\frac{1}{2}$

$2\frac{7}{12}$

Correct answer:

$2\frac{7}{12}$

Explanation:

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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Example Question #44 : Arithmetic

What is the area of a rectangle with $l =3\frac{5}{9}$ cm and a $w = 2 \frac{1}{4}$ cm?

Possible Answers:

$9cm^{2}$

$8\frac{1}{2}cm^{2}$

$8cm^{2}$

$5 \frac{6}{13}cm^{2}$

Correct answer:

$8cm^{2}$

Explanation:

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

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

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

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

Then multiply the numerators and the denominators. Simplify.

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

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:

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

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

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

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Example Question #51 : Hspt Mathematics

The gas tank in Renee's mustang is filled to capacity at $13\frac{1}{2}$ gallons. She used $\frac{2}{3}$ of a tank of gas to take a drive to the shore. How much gas did it take to drive to the shore?

Possible Answers:

gallons

$8 \frac{1}{2}$ gallons

$9 \frac{1}{2}$ gallons

Correct answer:

gallons

Explanation:

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

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

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

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

gallons of gas was consumed on Renee's trip to the shore.

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HSPT Math : Fractions

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #5 : How To Multiply And Divide Fractions

Example Question #43 : Arithmetic

Example Question #6 : How To Multiply And Divide Fractions

Example Question #43 : Arithmetic

Example Question #42 : Problem Solving

Example Question #45 : Problem Solving

Example Question #41 : Arithmetic

Example Question #45 : Problem Solving

Example Question #44 : Arithmetic

Example Question #51 : Hspt Mathematics

All HSPT Math Resources

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