Fractions - PSAT Math

Card 0 of 1526

Question

Simplify:

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?

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 $\dpi{100} \small \frac {x}{y}$ ?

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:

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

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

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

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

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

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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Question

Define an operation $\otimes ;$ as follows:

For all real numbers ,

$a \otimes b = \frac{4-a}{b}$ .

Evaluate $1 \frac{1}{2} \otimes 2 \frac{1}{3}$ .

Answer

$a \otimes b = \frac{4-a}{b}$ , or, equivalently,

$a \otimes b =( 4-a ) \div b$

$1 \frac{1}{2} \otimes 2 \frac{1}{3} =\left ( 4-1 \frac{1}{2} \right ) \div 2 \frac{1}{3}$

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

$= \frac{5}{2} \div \frac{7}{3}$

$= \frac{5}{2} \times \frac{3}{7}$

$= \frac{15}{14}$

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

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Question

Define an operation $\ddagger$ as follows:

For all real numbers ,

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ .

Evaluate $\frac{3}{4} \ddagger\frac{4}{3}$ .

Answer

$a \ddagger b = \frac{a}{b} - \frac{b}{a}$ ,

or, equivalently,

$a \ddagger b = (a \div b) - (b \div a)$

$\frac{3}{4} \ddagger\frac{4}{3} =\left (\frac{3}{4} \div \frac{4}{3} \right ) - \left (\frac{4}{3} \div \frac{3}{4} \right )$

$=\left (\frac{3}{4} \times \frac{3}{4} \right ) - \left (\frac{4}{3} \times \frac{4} {3}\right )$

$= \frac{9}{16} -\frac{16}{9}$

From here we need to find a common denominator.

$=\frac{9}{16}\bigg(\frac{9}{9}\bigg)- \frac{16}{9}\bigg(\frac{16}{16}\bigg)$

$= \frac{81}{144} -\frac{256}{144}$

$= -\frac{175}{144}$

$= -1 \frac{31}{144}$

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Question

Define an operation $\amalg$ as follows:

For all real numbers ,

$a \amalg b = 5 - \frac{a}{b}$ .

Evaluate $1 \frac{1}{2}; \amalg ; 2 \frac{1}{3}$ .

Answer

$a \amalg b = 5 - \frac{a}{b}$

or, equivalently,

$a \amalg b = 5 - a \div b$

$1 \frac{1}{2}; \amalg ; 2 \frac{1}{3}= 5 - 1 \frac{1}{2} \div 2 \frac{1}{3}$

$= 5 - \frac{3}{2} \div \frac{7}{3}$

$= 5 - \frac{3}{2} \times \frac{3}{7}$

$= 5 - \frac{9}{14}$

$= 4 \frac{5}{14}$

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Question

Simplify:

$\frac{1+5}{\frac{6}{3+1}}$

Answer

Begin by simplifying any additions that need to be done:

$\frac{1+5}{\frac{6}{3+1}}$

becomes

$\frac{6}{\frac{6}{4}}$

Now, remember that the numerator can be rewritten $\frac{6}{1}$ :

$\frac{\frac{6}{1}}{\frac{6}{4}}$

Now, when you divide fractions, you multiply the numerator by the reciprocal of the denominator:

$\frac{6}{1} * \frac{4}{6}$

Cancel the s and you get:

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Question

Write 7.5% as a fraction.

Answer

First convert the percentage to a decimal:

7.5% = .075

Then turn this into a fraction:

.075 = 75/1000

Simplify by dividing the numerator and denominator by 25:

75/1000 = 3/40

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Question

Write as a fraction: 22%

Answer

22% = 22/100

Divide everything by 2:

22/100 = 11/50

11 is a prime number, so this is as reduced as this fraction can get.

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Question

25% of 64 is equal to 5% of what number?

Answer

25% of 64 is 16 (you can find this with a calculator by 0.25 * 64). Divide 16 by 0.05 (or 1/20) to get the value of the number 16 is 5% of. (Or mental math of 16 * 20)

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Question

When y is decreased by ten percent, the result is equal to fifteen percent of x. Assuming both x and y are nonzero, what is the ratio of x to y?

Answer

The problem states that decreasing y by ten percent gives us the same thing as taking fifteen percent of x. We need to find an expression for decreasing y by ten percent, and an expression for fifteen percent of x, and then set these two things equal.

If we were to decrease y by ten percent, we would be left with ninety percent of y (because the percentages must add to one hundred percent). We could write ninety percent of y as 0.90_y_ = (90/100)y = (9/10)y. Remember, when converting from a percent to a decimal, we need to move the decimal two places to the left.

Similarly, we can write 15% of x as 0.15_x_ = (15/100)x = (3/20)x.

Now, we set these two expressions equal to one another.

(9/10)y = (3/20)x

Multiply both sides by 20 to eliminate fractions.

18_y_ = 3_x_

The question asks us to find the ratio of x to y, which is equal to x/y. Thus, we must rearrange the equation above until we have x/y by itself on one side.

18_y_ = 3_x_

Divide both sides by 3.

6_y_ = x

Divide both sides by y.

6 = x/y

Thus, the ratio of x to y is 6.

The answer is 6.

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Question

Convert 62% into simplified fraction form.

Answer

To convert a percent into a fraction, simply divide by 100 and reduce. In our case, 62 and 100 have the common factor of 2, so solving and simplification are as follows:

$\frac{62}{100}\rightarrow \frac{31}{50}$

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Question

If Mr. Jones’ math class has 8 boys and two-thirds of the class are girls, how many total students are in the class?

Answer

If two-thirds of the class are girls, then one-third must be boys. Set up an equation comparing the number of boys to how much they represent in the entire class:

8 = (1/3) x, where x is the number in the entire class.

When we solve for x in the equation we get x = 24.

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Question

Mr. Owens spent \$7.50 for a dinner buffet. The amount he paid accounted for 3/4 of the money in his wallet. How much money is left in his wallet for other expenses?

Answer

If \$7.50 is 3/4 of the total, 7.50/3 gives us what 1/4 of his total money would be. This equals \$2.50, the remaining unspent quarter.

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Question

A certain ball that is dropped will bounce back to 3/5 of the height it was initially dropped from. If after the 2nd bounce the ball reaches 39.96 ft, what was the initial height the ball was dropped from?

Answer

We know the height of the initial bounce, so work backwards to find the initial height. 39.96/0.6 = 66.6 = height of ball after first bounce

66.6/0.6 = 111 ft

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