Fractions - HSPT Math

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Question

For a party, Marco buys 4 boxes of cookies, each containing 10 cookies. Marco gives each of his guests 3 cookies, and then he eats 6 cookies himself. He now has 4 cookies left. How many guests did Marco give cookies to?

Answer

He begins with 40 and ends with 4, so 36 cookies total were eaten. Since he ate 6 himself, that means that his guests ate 30. Since each guest ate 3 cookies, guests.

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Question

Adam has 3 siblings. When his mother bakes a cake, each of the 4 children is given ¼ of the cake. Adam only finished 1/3 of his share of cake. If the original cake had 12 slices, how many slices did Adam eat?

Answer

Each of the children gets 1/4 of the cake. Taking 1/4 of 12 shows that each child gets 3 slices. If Adam finishes 1/3 of his share, then he has eaten one out of the 3 slices.

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Question

The time is now 11:17 AM. What time will it be in three hours and twenty-four minutes?

Answer

The time is 11:17 AM. In 24 minutes, the minutes will read , so it wil be 11:41 AM. Let's then add three hours to it.

The time will be 2:41 PM.

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Question

Steven purchased $1\frac{2}{3}:lbs$ of vegetables on Monday and $2\frac{3}{4}:lbs$ of vegetables on Tuesday. What was the total weight, in pounds, of vegetables purchased by Steven?

Answer

To solve this answer, we have to first make the mixed numbers improper fractions so that we can then find a common denominator. To make a mixed number into an improper fraction, you multiply the denominator by the whole number and add the result to the numerator. So, for the presented data:

$1\frac{2}{3}=\frac{(31)+2}{3}=\frac{3+2}{3}=\frac{5}{3}$

and

$2\frac{3}{4}=\frac{(42)+3}{4}=\frac{8+3}{4}=\frac{11}{4}$

Now, to find out how many total pounds of vegetables Steven purchased, we need to add these two improper fractions together:

$\frac{5}{3}+\frac{11}{4}$

To add these fractions, they need to have a common denominator. We can adjust each fraction to have a common denominator of by multiplying $\frac{5}{3}$ by $\frac{4}{4}$ and $\frac{11}{4}$ by $\frac{3}{3}$ :

$(\frac{5}{3}\frac{4}{4})+(\frac{11}{4}\frac{3}{3})$

To multiply fractions, just multiply across:

$\frac{20}{12}+\frac{33}{12}$

We can now add the numerators together; the denominator will stay the same:

$\frac{53}{12}$

Since all of the answer choices are mixed numbers, we now need to change our improper fraction answer into a mixed number answer. We can do this by dividing the numerator by the denominator and leaving the remainder as the numerator:

$53\div12=4: r.:5$

$\frac{53}{12}=4\frac{5}{12}$

This means that our final answer is $4\frac{5}{12}:lbs$ .

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Question

$-6\frac{1}{3} + 4\frac{1}{6}$

Answer

$-6\frac{1}{3} + 4\frac{1}{6}$

To solve, convert all mixed numbers to improper fractions:

$-6\frac{1}{3} = -\frac{(3\times 6) +1}{3} = -\frac{19}{3}$

$4\frac{1}{6} = \frac{(6\times 4)+1}{6} = \frac{25}{6}$

Because the denominators are not the same you have to find the Least Common Multiple, which is the Least Common Denominator of 3 and 6, which is 6. Then convert both fractions, if needed, to equivalent fractions with 6 as the denominator.

$-\frac{19}{3} =-\frac{38}{6}$ These are equivalent because the numerator and the denominator have been multiplied by 2 to get the equivalent fraction.

$-\frac{38}{6} +\frac{25}{6} = -\frac{13}{6} = -2\frac{1}{6}$

The answer is negative because the numerator with the greatest absolute value is -38 and that is a negative number.

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Question

$-\frac{1}{3} - (-\frac{3}{4})$

Answer

$-\frac{1}{3} - (-\frac{3}{4}) =$

$-\frac{1}{3} +\frac{3}{4}$ The subtraction of a negative changes the subtraction sign to a positive.

Find the LCD of 3 and 4 and convert to equivalent fractions, with the LCD as the denominator for both fractions.

The LCD is 12.

$-\frac{1}{3} = -\frac{4}{12}$ The numerator and the denominator were multiplied by 4.

$\frac{3}{4} = \frac{9}{12}$ The numerator and the denominator were multiplied by 3.

$-\frac{4}{12} + \frac{9}{12} = \frac{5}{12}$

Because the absolute value of 9 is greater than the absolute value of -4, the answer is positive.

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Question

$2\frac{3}{4} -6\frac{2}{3}$

Answer

$2\frac{3}{4} -6\frac{2}{3}$

First, convert all mixed numbers to improper fractions.

$2\frac{3}{4} = \frac{(4\times 2)+3}{4} =\frac{11}{4}$

$6\frac{2}{3} = \frac{(3\times 6)+2}{3} = \frac{20}{3}$

Because the denominators are not alike, the fractions have to be converted to equivalent fractions with the same denominator. This is done by finding the LCM which becomes the Least Common Denominator. The LCD of 4 and 3 is 12.

$\frac{11}{4} = \frac{33}{12}$ Both the numerator and the denominator were multiplied by 3.

$\frac{20}{3} = \frac{80}{12}$ Both the numerator and the denominator were multiplied by 4.

$\frac{33}{12} - \frac{80}{12} = \frac{33-80}{12} = -\frac{47}{12}$

The answer is negative because the method to solving would be to subtract and take the sign of the higher number.

$-\frac{47}{12} = -3\frac{11}{12}$

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Question

$2\frac{1}{8} + 1\frac{13}{16} + 1\frac{3}{4}$

Answer

Convert all mixed numbers to improper fractions.

$2\frac{1}{8}=\frac{(8\times 2)+1}{8} = \frac{17}{8}$ $2\frac{1}{8} =\frac{(8\times 2)+1}{8} =\frac{17}{8}$

$1\frac{13}{16} = \frac{(16\times 1)+13}{16} = \frac{29}{16}$

$1\frac{3}{4} =\frac{(4\times 1)+3}{4} = \frac{7}{4}$

Because all of the denominators are not the same find the LCM. This will become the LCD or Least Common Denominator.

Multiples of 8 are 8,16,24,32.

Multiples of 16 are 16, 32, 48.

Multiples of 4 are 4, 8, 12, 16.

The Least Common Multiple of 8,16 and 4 is 16. The LCD is 16.

$\frac{17}{8} =\frac{34}{16}$ The numerator and denominator were both multiplied by 2.

$\frac{7}{4} = \frac{28}{16}$ The numerator and the denominator were both multiplied by 4.

Now, that all of the fractions have the same denominator, add the numerators.

$\frac{34+29+28}{16} = \frac{91}{16} = 5\frac{11}{16}$

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Question

$-2\frac{2}{5} + (-3\frac{1}{3})$

Answer

$-2\frac{2}{5} + (-3\frac{1}{3})$

First, convert all mixed numbers to improper fractions.

$-2\frac{2}{5} = -\frac{(5\times 2)+2}{5} = -\frac{12}{5}$

$-3\frac{1}{3} = -\frac{(3\times 3)+1}{3} = -\frac{10}{3}$

Because the denominators are different, find the LCM Of 3 and 5. This becomes the LCD or least common denominator.

The LCD of 5 and 3 is 15.

Create equivalent fractions with 15 as the denominator.

$-\frac{12}{5} = -\frac{36}{15}$ Both numerator and denominator were multiplied by 3.

$-\frac{10}{3} = -\frac{50}{15}$ Both numerator and denominator were multiplied by 5.

$-\frac{36}{15} + -\frac{50}{15} = \frac{-36+-50}{15}$ When adding two negative integers, the result is negative.

$-\frac{86}{15} = -5\frac{11}{15}$

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Question

Evaluate if $a = \frac{5}{7}$ and $b = -\frac{2}{5}$ .

Answer

Evaluate if $a = \frac{5}{7}$ and $b = -\frac{2}{5}$

$\frac{5}{7} - (-\frac{2}{5}) =$

$\frac{5}{7} +\frac{2}{5}$ The subtraction of a negative changes the operation from subtraction to addition.

Find the Least Common Multiple of 7 and 5.

Multiples of 7 are 7, 14, 21, 28, 35.

Multiples of 5 are 5, 10, 15, 20 ,25, and 35.

You can also just multiply the denominators.

The LCM or the LCD of 7 and 5 is 35.

Create equivalent fractions with 35 as the denominator.

$\frac{5}{7} =\frac{25}{35}$ Both the numerator and denominator were multiplied by 5.

$\frac{2}{5} = \frac{14}{35}$ Both the numerator and denominator were multiplied by 7.

Now that the denominators are the same, just add the numerators.

$\frac{25}{35} +\frac{14}{35} = \frac{25+14}{35} = \frac{39}{35 } = 1\frac{4}{35}$

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Question

$14 -3\frac{1}{8}$

Answer

$14 -3\frac{1}{8}$

Change all whole numbers and mixed numbers to improper fractions.

$14 = \frac{14}{1}$ When there is a whole number, just place the number 1 underneath the fraction bar as the denominator.

$3\frac{1}{8} = \frac{(8\times 3)+1}{8} = \frac{25}{8}$

Because the denominators are different, create equivalent fractions by using the Least Common Denominator. The LCD of 8 and 1 is 8.

$\frac{14}{1} =\frac{112}{8}$

$\frac{25}{8}$ does not have to be changed, because the denominator is already the LCD, which is 8.

Now that the denominators are the same, subtract.

$\frac{112}{8} -\frac{25}{8} = \frac{112-25}{8} = \frac{87}{8}$

$\frac{87}{8} = 10\frac{7}{8}$

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Question

Evaluate if $x= -\frac{5}{8}$ and $y = 2\frac{5}{6}$ .

Answer

Evaluate if $x= -\frac{5}{8}$ and $y = 2\frac{5}{6}$ .

$-\frac{5}{8} -2\frac{5}{6}$

First, convert any mixed numbers to improper fractions.

$2\frac{5}{6} = \frac{(6\times 2)+5}{6} = \frac{17}{6}$

$-\frac{5}{8} - \frac{17}{6}$

In order to solve, find the common denominator of 8 and 6. This is found by listing the multiples and picking the LCM or the least common multiple. This becomes the LCD or Least Common Denominator.

Multiples of 8 are 8, 16, 24.

Multiples of 6 are 6, 12,18, and 24.

The LCM or the LCD will be 24.

Create equivalent fractions for both with 24 as the denominator,

$-\frac{5}{8} =-\frac{15}{24}$ Both numerator and denominator were multiplied by 3.

$-\frac{17}{6} = -\frac{68}{24}$ Both numerator and denominator were multiplied by 4.

$-\frac{15}{24} + (-\frac{68}{24}) = \frac{-15 + (-68)}{24} = -\frac{83}{24}$ When adding two negatives, the result is negative.

$-\frac{83}{24} = -3\frac{11}{24}$

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Question

$\frac{9}{4}-\frac{3}{2}=?$

Answer

To add or subtract fractions, you need a common denominator.

You can multiple the second fraction by two to get a common denominator of four.

$\frac{3}{2}\times\frac{2}{2}=\frac{6}{4}$

Then you just subtract the numerators to get

$\frac{9}{4}-\frac{6}{4}=\frac{3}{4}$ .

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Question

Find the sum of the following:

$5\frac{1}{8}+6\frac{3}{16}?$

Answer

To get the decimal from a fraction, divide the numerator by the denominator

$\frac{1}{8}=0.125$

$5\frac{1}{8}=5.125$

$\frac{3}{16}=0.1875$

$6\frac{3}{16}=6.1875$

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Question

What is a fraction form of .036?

Answer

The fraction form of .036 is $\dpi{100} \frac{36}{1000}$ . None of our answers match this fraction, so we must reduce it. The easiest way to do so is to divide the top and bottom by 2.

$\dpi{100} \frac{36}{1000}=\frac{18}{500}$

We can divide by 2 again:

$\dpi{100} \frac{18}{500}=\frac{9}{250}$

This is one of our answers (and also the fraction in its most reduced form).

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Question

$\dpi{100} 2\frac{1}{4}+5\frac{4}{5}=$

Answer

First add the whole numbers.

$\dpi{100} 2+5=7$

Then add the fractions. In order to this, the fractions need to have a common denominator. The denominators are 4 and 5, so we need to find the least common multiple of 4 and 5. We can do this by finding all the multiples of the larger number (5) and try to divide each of them by 4.

5: No

10: No

15: No

20: Yes

So the least common multiple is 20. We can multiply $\dpi{100} 4\times 5$ to get 20, and $\dpi{100} 5\times 4$ to get 20.

$\dpi{100} \frac{1}{4}+\frac{4}{5}=\frac{5\times 1}{5\times 4}+\frac{4\times 4}{4\times 5}=\frac{5}{20}+\frac{16}{20}=\frac{21}{20}$

We convert $\dpi{100} \frac{21}{20}$ to $\dpi{100} 1\frac{1}{20}$ and add it to 7 to get $\dpi{100} 8\frac{1}{20}$ .

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Question

$1 \frac{1}{2} + 2 \frac{3}{4} =$

Answer

I think the easiest way to do this problem is to make a common denominator. 4 is the smallest common denominator.

$2 \frac{3}{4}$ is equivalent to $\frac{11}{4}$ .

When you add $\frac{6}{4}$ and $\frac{11}{4}$ you get $\frac{17}{4}$ .

$\frac{17}{4}$ reduces to $4 \frac{1}{4}}$ and is equivalent to .

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Question

What is $\frac{5}{12}-\frac{3}8{}=?$

Answer

To subtract fractions we must first ensure that we have the same denominator which is the bottom part of the fraction.

To do this we must find the least common multiple of the denominators.

The least common multiple is the smallest number that multiples of both of the denominators multiply to. For example, for 2 and 6 it would be 6, for 2 and 5 it would be 10, for 3 and 8 it would be 24.

In this case the LCM is .

We then multiply the numerator and denominator of the fractions by the numbers it takes to reach the LCM $\frac{5}{12}\frac{2}{2}$ and $\frac{3}{8}\frac{3}{3}$ to get the same denominator because anything divided by itself is one so the fractions maintain their same value as the numbers change into the format we need to determine the answer.

To multiply a fraction you multiply the top number of the first fraction by the top number of the second fraction and the bottom number of the first fraction by the bottom number of the second fraction.

After doing this we then subtract the first numerator (top part of the fraction) from the second numerator and place the result over the new denominator

$\frac{10}{24}-\frac{9}{24}=\frac{1}{24}$

The final answer is $\frac{1}{24}$

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Question

Add the fractions below.

$\frac{4}{5} + \frac{7}{10}$

Answer

First, we need to find a common denominator. In this case, we can convert both denominators to .

$\frac{4}{5} + \frac{7}{10} = (\frac{4 }{5 }*\frac{2}{2}) + \frac{7}{10} = \frac{8}{10} + \frac{7}{10}$

Now we can add the numerators.

$\frac{8}{10}+\frac{7}{10}=\frac{15}{10}$

Both the numerator and denominator are divisible by , so we can simplify.

$\frac{15}{10}= \frac{15\div 5}{10\div 5}= \frac{3}{2}$

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Question

Find the arithmetic mean of the following five numbers.

$\frac{2}{5}, \frac{1}{5},\frac{1}{2},\frac{3}{10}, \frac{2}{5}$

Answer

Add the numbers, then divide by or, equivalently, multiply by $\frac{1}{5}$ .

$\frac{\frac{2}{5}+ \frac{1}{5}+\frac{1}{2}+\frac{3}{10}+ \frac{2}{5}}{5}$

$\frac{1}{5} \left(\frac{2}{5}+ \frac{1}{5}+\frac{1}{2}+\frac{3}{10}+ \frac{2}{5}\right )$

Find the common denominator of the terms we are adding.

$\frac{1}{5} \left(\frac{4}{10}+ \frac{2}{10}+\frac{5}{10}+\frac{3}{10}+ \frac{4}{10}\right )$

$\frac{1}{5} \left(\frac{4+2+5+3+4}{10} \right)$

Multiply and simplify.

$\frac{1}{5} \left(\frac{18}{10} \right)= \frac{1}{5} \left(\frac{9}{5} \right) = \frac{9}{25}$

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