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PSAT Math : Pattern Behaviors in Exponents

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Example Question #1 : Pattern Behaviors In Exponents

On January 15, 2015, Philip deposited $10,000 in a certificate of deposit that returned interest at an annual rate of 8.125%, compounded monthly. How much will his certificate of deposit be worth on January 15, 2020?

Possible Answers:

$\$15,007.02$

$\$ 14,991.24$

$\$15,011.78$

$\$14,778.51$

$\$14,062.50$

Correct answer:

$\$ 14,991.24$

Explanation:

The formula for compound interest is

$A = A_{0}\left ( 1+ \frac{r}{N} \right )^{Nt}$

where $A_{0}$ is the initial investment, is the interest rate expressed as the decimal equivalent, is the number of periods per year the interest is compounded, is the number of years, and is the final value of the investment.

Set $A_{0} = 10,000, r = 0.08125, N = 12$ (monthly = 12 periods), and t = 5 , and evaluate :

$A = 10,000 \left ( 1+ \frac{0.08125}{12} \right )^{12 \cdot 5}$

$A \approx 10,000 \left ( 1.0067708\right )^{60}$

$A \approx 10,000 \cdot 1.499124$

$A \approx 1 4,991.24$

The CD will be worth $14,991.24.

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Example Question #2 : How To Find Compound Interest

Money is deposited in corporate bonds which yield 6.735% annual interest compounded monthly, and which mature after ten years. Which of the following responses comes closest to the percent by which the value of bonds increases?

Possible Answers:

$90 \%$

$100 \%$

$85 \%$

$105 \%$

$95 \%$

Correct answer:

$95 \%$

Explanation:

The formula for compound interest is

$A = A_{0}\left ( 1+ \frac{r}{N} \right )^{Nt}$

In the given scenario, t = 10 , r = 0.06735 , and N = 12 (monthly); substitute:

$A = A_{0}\left ( 1+ \frac{0.06735}{12} \right )^{12 \cdot 10}$

$A \approx A_{0}\left ( 1+0.0056125 \right )^{12 0}$

$A \approx A_{0}\left ( 1 .0056125 \right )^{12 0}$

$A \approx A_{0} \cdot 1.9574$

This meas that the final value of the bonds is about 1.96 times their initial value, or, equivalently, 96% greater than their initial value. Of the given responses, 95% comes closest.

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Example Question #1 : How To Find Compound Interest

Donna wants to deposit money into a certificate of deposit so that in exactly ten years, her investment will be worth $100,000. The interest rate of the CD is 7.885%, compounded monthly.

What should Donna's initial investment be, at minimum?

Possible Answers:

$\$45,569.99$

$\$ 46,815.46$

More information is needed to answer the question.

$\$45,802.62$

$\$55,912.78$

Correct answer:

$\$45,569.99$

Explanation:

The formula for compound interest is

$A = A_{0}\left ( 1+ \frac{r}{N} \right )^{Nt}$

Set A = 100,000, r = 0.07885, N = 12 (monthly = 12 periods), and t = 10 , and evaluate $A_{0}$ :

$100,000 = A_{0} \left ( 1+ \frac{0.07885}{12} \right )^{12 \cdot 10}$

$100,000 \approx A_{0} \left ( 1+0.0065708\right )^{120}$

$100,000 \approx A_{0} \left ( 1 .0065708\right )^{120}$

$100,000 \approx A_{0} \cdot 2.1944$

$A_{0} \approx 100,000 \div 2.1944$

$A_{0} \approx 45,569.99$

The correct response is $45,569.99.

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Example Question #1 : How To Find Compound Interest

Tom invests $\$15$ , 000 in a savings account with an annual interest rate of $6\%$ . If his investment is compounded semiannually, how much interest does he earn after years?

Possible Answers:

$\$1390.91$

$\$1942.35$

$\$1758.45$

$\$1882.63$

$\$1841.31$

Correct answer:

$\$1882.63$

Explanation:

In order to find the interest earned, used the compound interest formula

$\text{Final Balance} = \text{Principal} \cdot (1+\frac{IR}{C})^{(\text{Time})(\text{C})}$

where represents the number of times the account is compounded each year, and represents the interest rate expressed as a decimal.

$=15,000\cdot(1+\frac{.06}{2})^{(2)(2)}$

$=15,000\cdot(1+.03)^{4}$

$=15,000\cdot(1.03)^{4}$

=16882.63

The account is worth $16882.63 after two years. Therefore Tom earns $1882.63 in interest.

16882.63-15000=1882.63

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Example Question #121 : Exponents

If a^x·a⁴ = a¹² and (b^y)³ = b¹⁵, what is the value of x - y?

Possible Answers:

-9

-4

-2

Correct answer:

Explanation:

Multiplying like bases means add the exponents, so x+4 = 12, or x = 8.

Raising a power to a power means multiply the exponents, so 3y = 15, or y = 5.

x - y = 8 - 5 = 3.

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Example Question #1 : How To Find Patterns In Exponents

If p and q are positive integrers and 27^p= 9^q, then what is the value of q in terms of p?

Possible Answers:

(3/2)p

(2/3)p

Correct answer:

(3/2)p

Explanation:

The first step is to express both sides of the equation with equal bases, in this case 3. The equation becomes 3^3p= 3^2q. So then 3p = 2q, and q = (3/2)p is our answer.

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Example Question #123 : Exponents

Simplify 27^2/3.

Possible Answers:

729

125

Correct answer:

Explanation:

27^2/3 is 27 squared and cube-rooted. We want to pick the easier operation first. Here that is the cube root. To see that, try both operations.

27^2/3 = (27²)^1/3 = 729^1/3 OR

27^2/3 = (27^1/3)² = 3²

Obviously 3² is much easier. Either 3² or 729^1/3 will give us the correct answer of 9, but with 3² it is readily apparent.

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Example Question #124 : Exponents

If and are integers and

$\left ( \frac{1}{3} \right )^{a}=27^{b}$

what is the value of $a\div b$ ?

Possible Answers:

$-\frac{1}{3}$

$\frac{1}{3}$

Correct answer:

Explanation:

To solve this problem, we will have to take the log of both sides to bring down our exponents. By doing this, we will get $\dpi{100} \small a\ast log\left (\frac{1}{3} \right )= b\ast log\left ( 27 \right )$ .

To solve for $\dpi{100} \small \frac{a}{b}$ we will have to divide both sides of our equation by $\dpi{100} \small log\frac{1}{3}$ to get $\dpi{100} \small \frac{a}{b}=\frac{log\left ( 27 \right )}{log\left ( \frac{1}{3} \right )}$ .

$\dpi{100} \small \frac{log\left ( 27 \right )}{log\left ( \frac{1}{3} \right )}$ will give you the answer of –3.

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Example Question #125 : Exponents

If $\log 2=0.301$ and $\log 3=0.477$ , then what is $\log 12$ ?

Possible Answers:

1.079

1.592

1.116

1.346

1.255

Correct answer:

1.079

Explanation:

We use two properties of logarithms:

$log(xy) = log (x) + log (y)$

$log(x^{n}) = nlog (x)$

So $\log 12=2 \log2+\log3$

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Example Question #1 : How To Find Patterns In Exponents

Evaluate:

$x^{-3}x^{6}$

Possible Answers:

$x^{3}$

$x^{6}$

$x^{-18}$

$x^{9}$

$x^{-3}$

Correct answer:

$x^{3}$

Explanation:

$x^{m}\ast x^{n} = x^{m + n}$ , here m=-3 and n=6 , hence -3+6=3 .

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PSAT Math : Pattern Behaviors in Exponents

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Pattern Behaviors In Exponents

Example Question #2 : How To Find Compound Interest

Example Question #1 : How To Find Compound Interest

Example Question #1 : How To Find Compound Interest

Example Question #121 : Exponents

Example Question #1 : How To Find Patterns In Exponents

Example Question #123 : Exponents

Example Question #124 : Exponents

Example Question #125 : Exponents

Example Question #1 : How To Find Patterns In Exponents

All PSAT Math Resources

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