Arithmetic - PSAT Math

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Question

If a rectangle's length decreases by fifteen percent, and its width decreases by twenty percent, then by what percent does the rectangle's area decrease?

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Answer

Let's call the original length and width of the rectangle $l_{1}$ and $w_{1}$ , respectively.

The initial area, $A_{1}$ , of the rectangle is equal to the product of the length and the width. We can represent this with the following equation:

$A_{1}=l_{1}\cdot w_{1}$

Next, let $l_{2}$ and $w_{2}$ represent the length and width, respectively, after they have been decreased. The final area will be equal to $A_{2}$ , which will be equal to the product of the final length and width.

$A_{2}=l_{2}\cdot w_{2}$

We are asked to find the change in the area, which essentially means we want to compare $A_{1}$ and $A_{2}$ . In order to do this, we will need to find an expression for $A_{2}$ in terms of $l_{1}$ and $w_{1}$ . We can rewrite $l_{2}$ and $w_{2}$ in terms of $l_{1}$ and $w_{1}$ .

First, we are told that the length is decreased by fifteen percent. We can think of the full length as 100% of the length. If we take away fifteen percent, we are left with 100 – 15, or 85% of the length. In other words, the final length is 85% of the original length. We can represent 85% as a decimal by moving the decimal two places to the left.

$l_{2}$ = 85% of $l_{1}$ = $0.85l_{1}$

Similarly, if we decrease the width by 20%, we are only left with 80% of the width.

$w_{2}$ = 80% of $w_{1}$ = $0.80w_{1}$

We can now express the final area in terms of $l_{1}$ and $w_{1}$ by substituting the expressions we just found for the final length and width.

$A_{2}=l_{2}\cdot w_{2}$ = ( $0.85l_{1}$ )( $0.80w_{1}$ ) = $0.68l_{1}w_{1}$

Lastly, let's apply the formula for percent of change, which will equal the change in the area divided by the original area. The change in the area is equal to the final area minus the original area.

percent change = $\frac{(A_{2}-A_{1})}{A_{1}}$ (100%)

= $\frac{(0.68l_{1}w_{1}-l_{1}w_{1})}{l_{1}w_{1}}$ (100%)

= $\frac{-0.32l_{1}w_{1}}{l_{1}w_{1}}$ (100%) = –0.32(100%) = –32%

The negative sign indicates that the rectangle's area decreased. The change in the area was a decrease of 32%.

The answer is 32.

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Question

Two two-digit numbers, and , sum to produce a three-digit number in which the second digit is equal to . The addition is represented below. (Note that the variables are used to represent individual digits; no multiplication is taking place).

What is ?

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Answer

Another way to represent this question is:

$\begin{matrix} & \ \ AB\ +& \ \ BA\ & 1A2 \end{matrix}$

In the one's column, and add to produce a number with a two in the one's place. In the ten's column, we can see that a one must carry in order to get a digit in the hundred's place. Together, we can combine these deductions to see that the sum of and must be twelve (a one in the ten's place and a two in the one's place).

In the one's column:

The one carries to the ten's column.

In the ten's column:

The three goes into the answer and the one carries to the hundred's place. The final answer is 132. From this, we can see that because .

Using this information, we can solve for .

$A+B=12\ \text{and}\ A=3$

You can check your answer by returning to the original addition and plugging in the values of and .

$AB=39\ \text{and}\ BA=93$

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Question

Simplify:

$\frac{x^2-y^2}{x+y}$

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Answer

_x_2 – _y_2 can be also expressed as (x + y)(x – y).

Therefore, the fraction now can be re-written as (x + y)(x – y)/(x + y).

This simplifies to (x – y).

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Question

Simplify the following expression:

$\frac{4x^6}{8x^3}$

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Answer

Following this equation, you divide 4 by 8 to get 1/2. When a variable is raised to an exponent, and you are dividing, you subtract the exponents, so 6 – 3 = 3.

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Question

Simplify the given fraction:

$\frac{125}{2000}$

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Answer

125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.

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Question

Simplify the given fraction:

$\frac{120}{6000}$

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Answer

120 goes into 6000 evenly 50 times, so we get 1/50 as our simplified fraction.

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Question

Let $\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}$ = $\frac{a}{b}$ , where and are both positive integers whose greatest common factor is one. What is the value of ?

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Answer

First we want to simplify the expression: $\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}$ .

One way to simplify this complex fraction is to find the least common multiple of all the denominators, i.e. the least common denominator (LCD). If we find this, then we can multiply every fraction by the LCD and thereby be left with only whole numbers. This will make more sense in a little bit.

The denominators we are dealing with are 2, 3, 4, 5, and 6. We want to find the smallest multiple that these numbers have in common. First, it will help us to notice that 6 is a multiple of both 2 and 3. Thus, if we find the least common multiple of 4, 5, and 6, it will automatically be a multiple of both 2 and 3. Let's list out the first several multiples of 4, 5, and 6.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60

6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

The smallest multiple that 4, 5, and 6 have in common is 60. Thus, the least common multiple of 4, 5, and 6 is 60. This also means that the least common multiple of 2, 3, 4, 5, and 6 is 60. Therefore, the LCD of all the fractions is 60.

Let's think of the expression we want to simplify as one big fraction. The numerator contains the fractions 1/4, 1/3, and –1/5. The denominator of the fraction is 1/2, –1/6 and 1. Remember that if we have a fraction, we can multiply the numerator and denominator by the same number without changing the value of the fraction. In other words, x/y = (xz)/(yz). This will help us because we can multiply the numerator (which consists of 1/4, 1/3, and –1/5) by 60, and then mutiply the denominator (which consists of 1/2, –1/6, and 1) by 60, thereby ridding us of fractions in the numerator and denominator. This process is shown below:

$\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}$ = $\frac{60}{60}\cdot$ $\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}$ = $\frac{60\cdot \frac{1}{4}+60\cdot \frac{1}{3}-60\cdot \frac{1}{5}}{60\cdot \frac{1}{2}-60\cdot \frac{1}{6}+60\cdot 1}$

= $\frac{15+20-12}{30-10+60}=\frac{23}{80}$

This means that a/b = 23/80. We are told that a and b are both positive and that their greatest common factor is 1. In other words, a/b must be the simplified form of 23/80. When a fraction is in simplest form, the greatest common factor of the numerator and denominator equals one. Since 23/80 is simplified, a = 23, and b = 80. The sum of a and b is thus 23 + 80 = 103.

The answer is 103.

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Question

Simplify:

(2_x_ + 4)/(x + 2)

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Answer

(2_x_ + 4)/(x + 2)

To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.

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Question

The expression $(\frac{a^{2}}{b^{3}})(\frac{a^{-2}}{b^{-3}}) = ?$

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Answer

A negative exponent in the numerator of a fraction can be rewritten with a positive exponent in the denominator. The same is true for a negative exponent in the denominator. Thus, $\frac{a^{-2}}{b^{-3}} =\frac{b^{3}}{a^{2}}$ .

When $\frac{a^{2}}{b^{3}}$ is multiplied by $\frac{b^{3}}{a^{2}}$ , the numerators and denominators cancel out, and you are left with 1.

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Question

A train travels at a constant rate of meters per second. How many kilometers does it travel in minutes? $(1\ km=1000\ m)$

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Answer

Set up the conversions as fractions and solve:

$\dpi{100} \small \frac{20m}{1sec}\times \frac{60sec^}{1min}\times \frac{1km}{1000m}\times \frac{10min}{1}$

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Question

Simplify. $\frac{4x^{4}z^{3}}{2xz^{2}}$

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Answer

To simplify exponents which are being divided, subtract the exponents on the bottom from exponents on the top. Remember that only exponents with the same bases can be simplified

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Question

Simplify the following expression:

$\frac{2x^{4}-32}{2x^{2}-8}$

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Answer

Factor both the numerator and the denominator:

$\frac{2(x^{2}-4)(x^{2}+4)}{2(x^{2}-4)}$

After reducing the fraction, all that remains is:

$(x^{2}+4)$

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Question

Simplify:

$\frac{x\sqrt{2}-2\sqrt{2}}{\sqrt{2x}+2}$

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Answer

Notice that the $\sqrt{2}$ term appears frequently. Let's try to factor that out:

$\frac{x\sqrt{2}-2\sqrt{2}}{\sqrt{2x}+2} = \frac{\sqrt{2}(x-2)}{\sqrt{2}(\sqrt{x} +\sqrt{2})} = \frac{x-2}{\sqrt{x} +\sqrt{2}}$

Now multiply both the numerator and denominator by the conjugate of the denominator:

$\frac{x-2}{\sqrt{x} +\sqrt{2}} * \frac{\sqrt{x} -\sqrt{2}}{\sqrt{x} -\sqrt{2}} =\frac{(x-2)*(\sqrt{x} -\sqrt{2})}{x-2} =\sqrt{x} -\sqrt{2}$

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Question

Simplify:

$\frac{8x^3y^2z^2}{4xy^4z^3}$

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Answer

$\frac{8x^3y^2z^2}{4xy^4z^3}$

To simply a fraction with variables, subtract exponents of like bases:

This leaves us with the expression:

$2x^2y^{-2}z^{-1}$

Next, we know that to change negative exponents in the numerator into positive exponents, we place them in the denominator. Thus, our expression simplifies to:

$\frac{2x^2}{y^2z}$

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Question

Reduce the fraction:

$\frac{12}{96}$

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Answer

The numerator and denominator are both divisible by 12. Thus, we divide both by 12 to get our final answer.

If we instead divide by another common factor, we may need to complete the process again to make sure that we have completely reduced the fraction.

In mathematical words we get the following:

$\frac{12}{96}=\frac{1 \cdot 12}{8 \cdot 12}=\frac{1}{8}$

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Question

Simplify the expression, $\left(\frac{x^2}{y^{-3}}\right)\left(\frac{x^{-1}}{y^3}\right)$ .

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Answer

The and variables with negative exponents can be rewritten with positive exponents by moving them from the denominator to the numerator, and vice versa. Therefore, the expression can be rewritten as

$\left(x^2y^3 \right) \left(\frac{1}{xy^3}\right) = \frac{x^2y^3}{xy^3}$ .

The exponents on the denominator can then be subtracted from the exponent in the numerator to give

$x^{2-1}y^{3-3} = x^1y^0 = x \cdot1 = x$

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Question

Simplify the expression $\left(\frac{2x^2}{ab^4}\right)(a^5b^2x^{-1})$ .

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Answer

The expression can be rewritten as

$\left(\frac{2x^2}{ab^4}\right)\left(\frac{a^5b^2}{x}\right)$

Now the expression can be combined by adding and subtracting exponents

$\left(\frac{2x^2a^5b^2}{ab^4x}\right) = 2x^{(2-1)}a^{(5-1)}b^{(2-4)} = 2xa^4b^{-2} = \frac{2xa^4}{b^2} = \frac{2a^4x}{b^2}$

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Question

Simplify the expression

$(12X^2Y^{-3}Z^2)\left(\frac{X^3Y^4Z^{-3}}{4}\right)$ .

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Answer

We can combine the expression into

$\left(\frac{12X^2X^3Y^{-3}Y^4Z^2Z^{-3}}{4}\right) = \frac{12}{4} X^{(2+3)}Y^{(-3+4)}Z^{(2-3)}$

The combined expression can then be simplified to

$3X^5Y^1Z^{-1} = \frac{3X^5Y}{Z}$

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Question

Simplify the expression

$\left(\frac{A^{\frac{3}{2}}\sqrt{B}}{C^{-\frac{1}{2}}}\right)\left(\frac{B^{-\frac{1}{2}}}{2\sqrt{AC}}\right)$ .

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Answer

The expression can be rewritten as

$\left(\frac{A^{\frac{3}{2}}B^\frac{1}{2}}{C^{-\frac{1}{2}}}\right)\left(\frac{B^{-\frac{1}{2}}}{2A^\frac{1}{2}C^\frac{1}{2}}\right)$

We can now move the variables to the numerator and combine alike variables

$(A^\frac{3}{2}B^\frac{1}{2}C^\frac{1}{2})\left(\frac{A^{-\frac{1}{2}}B^{-\frac{1}{2}}C^{-\frac{1}{2}}}{2}\right) = \frac{A^{(\frac{3}{2}-\frac{1}{2})}B^{(\frac{1}{2}-\frac{1}{2})}C^{(\frac{1}{2}-\frac{1}{2})}}{2}$

This becomes

$\frac{A^{\frac{2}{2}}B^0C^0}{2} = \frac{A^1}{2} = \frac{A}{2}$

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Question

Simplify the expression

$\frac{a^2-b^2}{a-b}$ .

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Answer

The numerator of this expression can be factored, giving the expression

$\frac{(a+b)(a-b)}{(a-b)}$

The will cancel itself out leaving,

$\frac{a+b\cdot1}{1} = a+b$

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