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

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

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?

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?

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?

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?

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

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

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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