Operations with Fractions - PSAT Math

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Question

Sally bought five computers for her office that cost $300, $405, $485, $520, and $555 respectively. She made a down payment of 2/5 the total cost and paid the rest in nine equal payments over the next nine months. Assuming no tax and no interest, what is the value of each of the nine payments?

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Answer

The total cost of the 5 computers is 2265.

2/5 of 2265 = 906, which is what Sally pays up front.

2265 – 906 = 1359, which is what Sally still owes.

1359/9 = 151, which is the value of each of the 9 equal payments.

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Question

The price of a computer is reduced by 1/8. The new price is then reduced by 1/6. What fraction of the original price is the current price?

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Answer

Let the original price = p.

After the first reduction, the price is (7/8)p

After the second reduction, the price is (5/6)(7/8)p = (35/48)p

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Question

If a car travels at 30 mph, how many feet per second does travel?

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Answer

30 miles / 1 hour * 5280 ft / 1 mile * 3600 seconds / 1 hour = 44 ft/sec

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Question

$\frac{5}{7}\frac{6}{4}\frac{12}{2}$

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Answer

Remember, when you multiply fractions, you can directly multiply their denominators and their numerators; therefore, you can begin this problem by making it into one large fraction:

$\frac{5}{7}\frac{6}{4}\frac{12}{2} = \frac{5612}{742}$

Now, you could multiply all of this out and then divide. However, you can cancel things immediately. The goes into the and the into the . Thus, you have:

$\frac{5612}{742} = \frac{533}{7}=\frac{9*5}{7}=\frac{45}{7}$

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Question

Simplify:

$\frac{15}{8}\frac{12}{5}\frac{1}{3}*14$

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Answer

First, begin by remembering that is the same as $\frac{14}{1}$ :

$\frac{15}{8}\frac{12}{5}\frac{1}{3}\frac{14}{1}$

Next, recall that you multiply fractions by multiplying the numerators and denominators by each other. It is very simple. This would give you:

$\frac{15}{8}\frac{12}{5}\frac{1}{3}\frac{14}{1}=\frac{1512114}{853}$

Now, you can cancel the and the :

$\frac{1512114}{853} = \frac{1214}{8}$

Next, you can reduce the and the :

$\frac{314}{2}$

You can also reduce the resulting and the :

$\frac{3*7}{1}=21$

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Question

In a group of 20 children, 25% are girls. How many boys are there?

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Answer

Since $\dpi{100} \small \frac{1}{4}$ of the children are girls, this totals to $\dpi{100} \small 20 \times \frac{1}{4} = 5$ girls in the group.

$\dpi{100} \small 20-5=15$ boys.

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Question

Jesse has a large movie collection containing X movies. 1/3 of his movies are action movies, 3/5 of the remainder are comedies, and the rest are historical movies. How many historical movies does Jesse own?

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Answer

1/3 of the movies are action movies. 3/5 of 2/3 of the movies are comedies, or (3/5)*(2/3), or 6/15. Combining the comedies and the action movies (1/3 or 5/15), we get 11/15 of the movies being either action or comedy. Thus, 4/15 of the movies remain and all of them have to be historical.

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Question

If x = 1/3 and y = 1/2, find the value of 2_x_ + 3_y_.

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Answer

Substitute the values of x and y into the given expression:

2(1/3) + 3(1/2)

= 2/3 + 3/2

= 4/6 + 9/6

= 13/6

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Question

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Answer

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Question

Solve $\frac{3}{7}+\frac{5}{8}-\frac{1}{2}$ .

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Answer

Finding the common denominator yields $\frac{24}{56}+\frac{35}{56}-\frac{28}{56}$ . We can then evaluate leaving $\frac{31}{56}$ .

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Question

What is the solution, reduced to its simplest form, for $x = \frac{7}{9}+\frac{3}{5}+\frac{2}{15}+\frac{7}{45}}$ ?

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Answer

$x=\frac{7}{9}+\frac{3}{5}+\frac{2}{15}+\frac{7}{45}=\frac{35}{45}+\frac{27}{45}+\frac{6}{45}+\frac{7}{45}=\frac{75}{45}=\frac{5}{3}$

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Question

What is the sum of $\small \frac{4}{5}$ and $\small 2 \frac{5}{6}$ ?

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Answer

We can begin by eliminating the obviously wrong answers. We know that the sum of the two fractions will be more than 1, so the answer choices $\tiny \tiny \frac{49}{109}$ and $\small \frac{30}{109}$ are out. Now, let's add the two fractions:

Begin by converting $\small 2 \frac{5}{6}$ to $\tiny \frac{17}{6}$ .

Now find the common denominator of $\small \frac{4}{5}$ and $\tiny \frac{17}{6}$ . The least common multiple of 5 and 6 is 30, so 30 is the common denominator. Now alter both fractions so that they use the common denominator:

$\tiny \small \frac{4}{5}= \frac{46}{56}=\frac{24}{30}$

$\tiny \small \frac{17}{6}=\frac{175}{65}=\frac{85}{30}$

Now we can easily add the two fractions together:

$\small \frac{24}{30}+\frac{85}{30}=\frac{109}{30}$

The answer is $\small \frac{109}{30}$ .

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Question

Simplify:

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Answer

Division is the same as multiplying by the reciprocal. Thus, a/b ÷ c/d = a/b x d/c = ad/bc

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Question

If p is a positive integer, and 4 is the remainder when p-8 is divided by 5, which of the following could be the value of p?

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Answer

Remember that if x has a remainder of 4 when divided by 5, xminus 4 must be divisible by 5. We are therefore looking for a number p such that p - 8 - 4 is divisible by 5. The only answer choice that fits this description is 17.

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Question

If $\dpi{100} \small x=\frac{2}{3}$ and $\dpi{100} \small y= \frac {3}{4}$ , then what is the value of $\dpi{100} \small \frac {x}{y}$ ?

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Answer

Dividing by a number (in this case $\dpi{100} \small \frac {3}{4}$ ) is equivalent to multiplying by its reciprocal (in this case $\dpi{100} \small \frac {4}{3}$ ). Therefore:

$\dpi{100} \small \frac {2}{3}\div \frac{3}{4} = \frac{2}{3}\times \frac{4}{3} = \frac{8}{9}$

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Question

Evaluate the following:

$\left ( \frac{5}{6}\times \frac{12}{13} \right )^{2}\div \frac{3}{4}$

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Answer

$\left ( \frac{5}{6}\times \frac{12}{13} \right )^{2}\div \frac{3}{4}$

First we will evaluate the terms in the parentheses:

$\left ( \frac{5}{6}\times \frac{2\times 6}{13} \right )^{2}\div \frac{3}{4}$

$\left ( \frac{5}{1}\times \frac{2}{13} \right )^{2}\div \frac{3}{4}$

$\left ( \frac{10}{13} \right )^{2}\div \frac{3}{4}$

Next, we will square the first fraction:

$\left ( \frac{10^{2}}{13^{2}} \right )\div \frac{3}{4}$

$\frac{100}{169}\div \frac{3}{4}$

We can evaluate the division as such:

$\frac{100}{169}\times\frac{4}{3}=\frac{400}{507}$

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Question

Simplify: $\frac{\frac{3x}{2}}{\frac{5x}{4}}$

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Answer

Start by rewriting this fraction as a division problem:

$\frac{3x}{2}\div \frac{5x}{4}$

When dividing fractions, you multiply by the reciprocal of the second fraction, so you can rewrite your problem like this:

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

Multiply across the numerators and then across the denominators to get $\frac{12x}{10x}$ . The x's cancel, and you can reduce the fraction to be $\frac{6}{5}$ .

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Question

Define an operation $\blacklozenge$ as follows:

For all real numbers ,

$a ; \blacklozenge ; b = a \div b^{2}$ .

Evaluate $25 ; \blacklozenge ; 6\frac{1}{4}$ .

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Answer

$a ; \blacklozenge ; b = a \div b^{2}$

$25 ; \blacklozenge ; 6\frac{1}{4}= 25 \div \left ( 6\frac{1}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{25}{4} \right ) ^{2}$

$= 25 \div \left ( \frac{625}{16} \right )$

$= \frac{25 }{1}\times \left ( \frac{16} {625}\right )$

$= \frac{1}{1}\times \left ( \frac{16} {25}\right )$

$= \frac{16} {25}$

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Question

Define an operation $\odot$ as follows:

For all real numbers ,

$a \odot b = \frac{a}{b+2}$ .

Evaluate $\frac{3}{5} \odot \frac{1}{5}$ .

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Answer

$a \odot b = \frac{a}{b+2}$ ,

or, equivalently,

$a \odot b = a \div\left ( b+2 \right )$

$\frac{3}{5} \odot \frac{1}{5}= \frac{3}{5} \div \left (\frac{1}{5}+2 \right )$

$= \frac{3}{5} \div \frac{11}{5}$

$= \frac{3}{5} \cdot \frac{5}{11}$

$= \frac{3}{1} \cdot \frac{1}{11}$

$= \frac{3} {11}$

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Question

Define an operation $\ominus$ as follows:

For all real numbers ,

$a ; \ominus ; b = a^{2} \div b$ .

Evaluate $16 ; \ominus ; 3\frac{1}{5}$ .

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Answer

$a ; \ominus ; b = a^{2} \div b$

$16 ; \ominus ; 3\frac{1}{5} = 16^{2} \div 3\frac{1}{5}$

$= 256 \div \frac{16}{5}$

$= \frac{256 }{1}\times \frac{5}{16}$

$= \frac{16}{1}\times \frac{5}{1}$

$= 16 \times 5 = 80$

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