SAT Mathematics - SAT Math

Card 0 of 20

Question

Add: $\frac{\frac{9}{8}}{7}+\frac{7}{\frac{1}{7}}$

Answer

To add $\frac{\frac{9}{8}}{7}+\frac{7}{\frac{1}{7}}$ , first simply each term by rewriting the terms using a division sign.

$\frac{9}{8}\div7 + 7 \div \frac{1}{7}$

Take the reciprocal of the terms after the division sign, and change the division sign to a multiplication sign. Simplify.

$\frac{9}{8}\times \frac{1}{7}+ 7 \times 7 = \frac{9}{56} + 49= 49\frac{9}{56}$

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Question

Add: $\frac{1}{\frac{3}{2}}+ \frac{\frac{4}{1}}{\frac{1}{5}}$

Answer

The terms shown are complex fractions. We first need to simplify each and find the least common denominator before solving.

Rewrite the complex fraction using a division sign.

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

Change the division sign to a multiplication sign, and take the reciprocal of the second term.

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

Repeat this step for the second term.

$\frac{\frac{4}{1}}{\frac{1}{5}} = \frac{4}{1} \div \frac{1}{5} = \frac{4}{1} \times \frac{5}{1}=20$

Add the two terms together.

The answer is: $20\frac{2}{3}$

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Question

Simplify: $\sqrt{-3}+\sqrt{-9}+\sqrt{-16}$

Answer

Rewrite $\sqrt{-3}+\sqrt{-9}+\sqrt{-16}$ in their imaginary terms.

$i=\sqrt{-1}$

$\sqrt{-3}+\sqrt{-9}+\sqrt{-16}=i\sqrt3+3i+4i = i\sqrt3 +7i$

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Question

For $i=\sqrt{-1}$ , what is the sum of $\frac{3}{2} + \frac{1}{4} i$ and its complex conjugate?

Answer

The complex conjugate of a complex number is , so $\frac{3}{2} + \frac{1}{4} i$ has $\frac{3}{2} - \frac{1}{4} i$ as its complex conjugate. The sum of the two numbers is

$\left (\frac{3}{2} + \frac{1}{4} i \right ) + \left ( \frac{3}{2} - \frac{1}{4} i \right )$

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

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

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Question

Add $5 + i \sqrt {5 }$ and its complex conjugate.

Answer

The complex conjugate of a complex number is . Therefore, the complex conjugate of $5 + i \sqrt {5 }$ is $5 - i \sqrt {5 }$ ; add them by adding real parts and adding imaginary parts, as follows:

$(5 + i \sqrt {5 } )+( 5 - i \sqrt {5 })$

$= 5 + 5 + i \sqrt {5 } - i \sqrt {5 }$

,

the correct response.

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Question

Add $13 - i \sqrt{13}$ to its complex conjugate.

Answer

The complex conjugate of a complex number is . Therefore, the complex conjugate of $13 - i \sqrt{13}$ is $13 + i \sqrt{13}$ ; add them by adding real parts and adding imaginary parts, as follows:

$(13 - i \sqrt{13} )+( 13 + i \sqrt{13})$

$(13 - i \sqrt{13} )+( 13 + i \sqrt{13})$

$= 13+13 - i \sqrt {13 }+ i \sqrt {13}$

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Question

An arithmetic sequence begins as follows:

Give the next term of the sequence

Answer

The common difference of an arithmetic sequence can be found by subtracting the first term from the second:

$d = a_{2} - a_{1}$

Add this to the second term to obtain the desired third term:

$a_{3} = a_{2} +d = 3i + ( -3 + 3i) = -3 + 3i + 3i = -3 + 6i$ .

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Question

Evaluate:

$i^{6} + i^{7}+ i^{8}+ i^{9}$

Answer

A power of can be evaluated by dividing the exponent by 4 and noting the remainder. The power is determined according to the following table:

$\begin{matrix} \textrm{\underline{Rem}}& \textrm{\underline{Power}} \ 0& 1\ 1& i\ 2& -1\ 3& -i \end{matrix}$

$6 \div 4 = 1 \textrm{ R }2$ , so $i^{6} = -1$

$7 \div 4 = 1 \textrm{ R }3$ , so $i^{7} = -i$

$8 \div 4 = 2 \textrm{ R }0$ , so $i^{8} = 1$

$9 \div 4 = 2 \textrm{ R }1$ , so $i^{9} = i$

Substituting:

$i^{6} + i^{7}+ i^{8}+ i^{9}$

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Question

Evaluate:

$9i^{9}+ 10 i^{10}+ 11 i^{11}+ 12 i^{12}$

Answer

A power of can be evaluated by dividing the exponent by 4 and noting the remainder. The power is determined according to the following table:

$\begin{matrix} \textrm{\underline{Rem}}& \textrm{\underline{Power}} \ 0& 1\ 1& i\ 2& -1\ 3& -i \end{matrix}$

$9 \div 4 = 2 \textrm{ R }1$ , so $i^{9} = i$

$10 \div 4 = 2 \textrm{ R }2$ , so $i^{10} = -1$

$11 \div 4 = 2 \textrm{ R }3$ , so $i^{11} = - i$

$12 \div 4 = 3 \textrm{ R }0$ , so $i^{12} = 1$

Substituting:

$9i^{9}+ 10 i^{10}+ 11 i^{11}+ 12 i^{12}$

Collect real and imaginary terms:

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Question

Simplify:

Answer

It can be easier to line real and imaginary parts vertically to keep things organized, but in essence, combine like terms (where 'like' here means real or imaginary):

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Question

If Johnny buys two comic books, priced at $1.50 each, and a candy bar, priced at $0.75, he'll have three quarters and two dimes left over. How much money does he have right now?

Answer

Add what he can purchase to what he has left over:

Two comic books and the candy bar: $1.50 + $1.0 + $0.75 = $3.75

Three quarters and two dimes: $0.75 + $0.20 = $0.95

Therefore his total amount of money is $3.75 + $0.95 = $4.70.

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Question

Add:

Answer

In order to add the decimals, add placeholders to the decimal .

Be careful not to add the wrong digits!

Add the thousandths places.

Add the hundredths places.

Add the tenths places.

Combine the numbers and put a decimal before the tenths place.

The correct answer is:

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Question

A family is taking a trip from Town A to Town B, then to Town C. Above is a diagram of the routes available to them. How many routes will only require them to drive 40 miles or fewer:

Answer

There are three routes from Point A to Point B, and three from Point B to Point C, for a total of $3 \times 3 = 9$ routes total. The total distance traveled is the distance of one of the first three routes added to that of one of the last three; we can take all nine possibilities and add the distances:

Five of these routes require driving a distance 40 miles or fewer.

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Question

A family is taking a trip from Town A to Town B, then to Town C. Above is a diagram of the routes available to them.

Give the range for the driving distance for the trip.

Answer

Each route includes one path from Point A to Point B and one path from Point B o Point C.

The shortest possible drive is the sum of the shortest paths for each leg of the trip:

The longest possible drive is the sum of the longest paths for each leg of the trip:

The correct response is that .

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Question

Above is a simplified map of the routes from Town A to Town B, and the routes from Town B to Town C.

A family wants to travel from Town A to Town C by way of Town B, then back to Town A by way of Town B. Since all routes are scenic, the family does not want to take any route twice.

Give the range for the distance in miles that the family will travel.

Answer

The family's trip will be designed so that the family will take two different routes of the three that connect Town A and Town B, and two different routes of the three that connect Town B and Town C.

The minimum distance that the family will travel is therefore the sum of the lengths of the two shortest routes from Town A to Town B, and those of the two shortest routes from Town B to Town C:

miles

The maximum distance that the family will travel is, similarly, the sum of the lengths of the two longest routes from Town A to Town B, and those of the two longest routes from Town B to Town C:

miles

The correct choice is therefore .

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Question

If x represents an even integer, which of the following expressions represents an odd integer?

Answer

Pick any even integer (2, 4, 6, etc.) to represent x. The only value that is odd is 3_x_ + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3_x_ + 1 is the only choice that adds an odd number to the product.

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Question

Add:

Answer

Add the ones digit.

Since the number is 10 or greater, the tens digit is the carryover.

Add the tens digit with the carryover.

Add the hundreds digit with the carryover.

Combine all the ones digits in our calculations.

The answer is .

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Question

If $y = 5^{a} * 5^{b}$ and , what is the value of ?

Answer

Multiplying two exponents that have the same base is the equivalent of simply adding the exponents.

So $y = 5^{a} * 5^{b}$ is the same as $y = 5^{a + b}$ , and if , then $y = 5^{3}$ or .

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Question

If $3^{4x+6}=27^{2x}$ , what is the value of ?

Answer

Using exponents, 27 is equal to 33. So, the equation can be rewritten:

34_x_ + 6 = (33)2_x_

34_x_ + 6 = 36_x_

When both side of an equation have the same base, the exponents must be equal. Thus:

4_x_ + 6 = 6_x_

6 = 2_x_

x = 3

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Question

If _a_2 = 35 and _b_2 = 52 then _a_4 + _b_6 = ?

Answer

_a_4 = _a_2 * _a_2 and _b_6= _b_2 * _b_2 * _b_2

Therefore _a_4 + _b_6 = 35 * 35 + 52 * 52 * 52 = 1,225 + 140,608 = 141,833

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