All practice questions

Question 1Question →

A rectangular tabletop measuring $126\text{ centimeters}$ by $180\text{ centimeters}$ is to be completely covered by non-overlapping, identical square tiles of the maximum possible side length, such that no tiles need to be cut. How many such square tiles are required to cover the entire tabletop?

Show answer & explanation

Answer: 70

Answer

70 tiles are required to cover the tabletop.

To cover a

126\text{ cm} \times 180\text{ cm}

surface with identical square tiles of maximum side length without cutting, the side length of each square tile must be the greatest common divisor of

126

and

180

, which is

18\text{ cm}

. Dividing the surface dimensions by

18\text{ cm}

gives

7

tiles along one side and

10

tiles along the other, yielding a total of

70

tiles.

Step-by-Step Solution

1

Determine the prime factorizations of the tabletop dimensions 126 and 180.

126 = 2^1 \times 3^2 \times 7^1

and

180 = 2^2 \times 3^2 \times 5^1

Prime factorization allows systematic extraction of the greatest common divisor.

2

Calculate the Greatest Common Divisor (GCD) of 126 and 180.

\text{GCD}(126, 180) = 2^{\min(1,2)} \times 3^{\min(2,2)} = 2^1 \times 3^2 = 18

The maximum side length of a square tile that fits evenly without cutting is the GCD of the two dimensions.

3

Find the total number of tiles required.

\text{Total tiles} = \left(\frac{126}{18}\right) \times \left(\frac{180}{18}\right) = 7 \times 10 = 70

The total area divided by the area of one tile gives the number of tiles needed.

Key Concept

Greatest Common Divisor (GCD) application in geometric spatial partitioning

Question 2Question →

Set $S$ consists of all consecutive integers from $-m$ to $n$ , inclusive, where $m$ and $n$ are positive integers with $n > m$ . If set $S$ contains exactly $25$ integers and the sum of all integers in set $S$ is $75$ , what is the value of $m$ ?

Show answer & explanation

Answer: 9

Answer

The value of

m

is

9

.

To find

m

, use the two fundamental properties of consecutive integer sets: term count and set sum. The number of terms from

-m

to

n

inclusive is

n - (-m) + 1 = n + m + 1 = 25

, which yields

n + m = 24

. The sum of an evenly spaced set is the product of the number of terms and the arithmetic mean of the smallest and largest terms:

25 \times \frac{-m + n}{2} = 75

, which simplifies to

n - m = 6

. Subtracting

n - m = 6

from

n + m = 24

gives

2m = 18

, so

m = 9

.

Step-by-Step Solution

1

Set up the equation for the number of terms in the set.

n + m = 24

The number of integers from

-m

to

n

inclusive is

n - (-m) + 1 = n + m + 1 = 25

.

2

Set up the equation for the sum of the integers in the set.

n - m = 6

The sum of an arithmetic progression is given by

\text{number of terms} \times \text{mean} = 25 \times \frac{-m + n}{2} = 75

, leading to

\frac{n - m}{2} = 3

.

3

Solve for

m

using the two linear equations.

m = 9

Subtracting

n - m = 6

from

n + m = 24

yields

2m = 18

, giving

m = 9

.

Key Concept

Properties of consecutive integer sets: inclusive term counting (

n - \text{start} + 1

) and set sum calculation (

\text{count} \times \text{mean}

).

Estimated Time:1m 30s

Question 3Question →

A fuel tank initially contains $60$ gallons of a fuel mixture that is $10\%$ ethanol and $90\%$ gasoline by volume. A mechanic removes $x$ gallons of this mixture and replaces it with an equal volume of pure ethanol to obtain a new mixture that is $25\%$ ethanol by volume. What is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The volume of fuel mixture that must be removed and replaced with pure ethanol is

10

gallons.

Replacing

10

gallons of the

10\%

ethanol fuel with pure ethanol removes

1

gallon of ethanol and adds

10

gallons of pure ethanol. The total ethanol in the tank becomes

6 - 1 + 10 = 15

gallons, which represents exactly

25\%

of the total

60

-gallon volume.

Step-by-Step Solution

1

Calculate the initial volume of ethanol in the tank

Ethanol volume =

0.10 \times 60 = 6

gallons.

Establishes the starting quantity of the solute.

2

Express the amount of ethanol after removal and replacement in terms of

x

Final ethanol volume =

6 - 0.10x + x = 6 + 0.90x

gallons.

Removing

x

gallons of fuel removes

10\%

ethanol, while adding

x

gallons of pure ethanol adds

100\%

ethanol.

3

Set up an equation using the target ethanol concentration

6 + 0.90x = 0.25 \times 60 = 15

.

The final

60

-gallon mixture must contain

25\%

ethanol by volume.

4

Solve the equation for

x

0.90x = 9 \implies x = 10

.

Isolates

x

to find the required replacement volume.

Key Concept

Dilution and fluid replacement in mixture problems

Question 4Question →

If $x$ and $y$ are integers such that $-5 \le x \le -1$ and $2 \le y \le 6$ , what is the minimum possible value of $\frac{x - y}{x + y}$ ?

Show answer & explanation

Answer: -11

Answer

The minimum possible value of the expression is -11.

The correct answer is -11 because the numerator

x - y

is negative for all allowed values of

x

and

y

. To minimize a negative fraction, the denominator

x + y

must be positive and minimized (equal to 1), while the numerator must be as negative as possible. Choosing

x = -5

and

y = 6

gives a numerator of

-11

and a denominator of

1

, producing the minimum value of -11.

Step-by-Step Solution

1

Evaluate the sign of the numerator

Since

x

is negative (

x \le -1

) and

y

is positive (

y \ge 2

),

x - y

is always negative (

x - y \le -3

).

Subtracting a positive integer from a negative integer yields a negative result.

2

Determine the conditions for minimizing a negative fraction

To obtain the minimum (most negative) value, the denominator

x + y

must be a positive integer and as small as possible.

A negative numerator divided by a positive denominator yields a negative quotient. Dividing by a smaller positive number yields a quotient with larger absolute value, making it smaller (more negative).

3

Identify the smallest positive denominator and solve for variables

The smallest positive integer for

x + y

is

1

. Setting

y = 1 - x

gives

x - y = 2x - 1

.

Since

x

and

y

are integers,

x + y

must be an integer.

4

Maximize the magnitude of the negative numerator

Using the lowest boundary

x = -5

gives

y = 6

. The value of the expression is

\frac{-5 - 6}{-5 + 6} = -11

.

Evaluating at

x = -5

and

y = 6

gives numerator

-11

and denominator

1

, resulting in

-11

.

Key Concept

Minimizing algebraic fractions involving signed numbers

Question 5Question →

Let $m = 2^a \cdot 5^3 \cdot 11^b$ and $n = 2^4 \cdot 5^c \cdot 11^2$ , where $a$ , $b$ , and $c$ are positive integers. If the greatest common divisor of $m$ and $n$ is $2^3 \cdot 5^2 \cdot 11^2$ , and the least common multiple of $m$ and $n$ has exactly $180$ positive integer divisors, what is the value of $a + b + c$ ?

Show answer & explanation

Answer: 13

Answer

The value of

a + b + c

is

13

.

To find

a, b, c

, analyze prime factor exponents. The GCD uses minimum exponents:

\min(a,4)=3 \implies a=3

,

\min(3,c)=2 \implies c=2

, and

\min(b,2)=2 \implies b \ge 2

. The LCM uses maximum exponents:

2^4, 5^3, 11^b

. The number of divisors of the LCM is

(4+1)(3+1)(b+1) = 20(b+1) = 180

, yielding

b=8

. Adding

a+b+c = 3+8+2 = 13

.

Step-by-Step Solution

1

Use the prime exponent rule for greatest common divisor to find

a

and

c

, and bound

b

.

a = 3

,

c = 2

, and

b \ge 2

.

The greatest common divisor takes the minimum exponent for each prime factor:

\min(a, 4) = 3 \implies a = 3

,

\min(3, c) = 2 \implies c = 2

, and

\min(b, 2) = 2 \implies b \ge 2

.

2

Determine the prime factorization of

\text{lcm}(m, n)

.

\text{lcm}(m, n) = 2^4 \cdot 5^3 \cdot 11^b

.

The least common multiple takes the maximum exponent for each prime factor:

\max(3, 4) = 4

,

\max(3, 2) = 3

, and

\max(b, 2) = b

since

b \ge 2

.

3

Apply the total number of divisors formula to solve for

b

.

b = 8

.

The number of positive integer divisors of

2^4 \cdot 5^3 \cdot 11^b

is

(4+1)(3+1)(b+1) = 20(b+1) = 180

, which yields

b+1 = 9

so

b = 8

.

4

Calculate the sum

a + b + c

.

a + b + c = 13

.

Summing the values

a=3

,

b=8

, and

c=2

gives

3 + 8 + 2 = 13

.

Key Concept

GCD and LCM prime factor exponent rules and total divisor formula

Question 6Question →

What is the remainder when the expression $7^{45} \cdot 3^{22} + 8^{15}$ is divided by $10$ ?

Show answer & explanation

Answer: 5

Answer

The remainder is 5.

Dividing any number by 10 leaves a remainder equal to the units digit of that number. By analyzing the units digit pattern (cyclicity of period 4) for powers of 7, 3, and 8:

7^{45} \equiv 7^1 = 7 \pmod{10}

,

3^{22} \equiv 3^2 = 9 \pmod{10}

, and

8^{15} \equiv 8^3 = 2 \pmod{10}

. The expression simplifies to

(7 \times 9) + 2 = 63 + 2 = 65

, which has a units digit of 5. Therefore, the remainder when divided by 10 is 5.

Step-by-Step Solution

1

Relate remainder modulo 10 to units digit cyclicity.

Finding the remainder when an expression is divided by 10 is equivalent to finding its units digit.

Any positive integer

N

can be expressed as

10k + r

, where

r

is the units digit and the remainder when

N

is divided by 10.

2

Determine the units digit of

7^{45}

.

7^{45} \equiv 7 \pmod{10}

.

Powers of 7 repeat their units digits in a cycle of length 4 (7, 9, 3, 1). Dividing the exponent 45 by 4 gives a remainder of 1, so

7^{45}

has the same units digit as

7^1 = 7

.

3

Determine the units digit of

3^{22}

.

3^{22} \equiv 9 \pmod{10}

.

Powers of 3 repeat their units digits in a cycle of length 4 (3, 9, 7, 1). Dividing the exponent 22 by 4 gives a remainder of 2, so

3^{22}

has the same units digit as

3^2 = 9

.

4

Determine the units digit of the product

7^{45} \cdot 3^{22}

.

7^{45} \cdot 3^{22} \equiv 3 \pmod{10}

.

The product of the units digits is

7 \times 9 = 63

, which has a units digit of 3.

5

Determine the units digit of

8^{15}

.

8^{15} \equiv 2 \pmod{10}

.

Powers of 8 repeat their units digits in a cycle of length 4 (8, 4, 2, 6). Dividing the exponent 15 by 4 gives a remainder of 3, so

8^{15}

has the same units digit as

8^3 = 512

, which ends in 2.

6

Combine the results to find the final remainder modulo 10.

5

Adding the units digit of the first term (3) and the second term (2) gives

3 + 2 = 5

.

Key Concept

Units Digit Cyclicity and Modular Arithmetic

Question 7Question →

If $5^{x} - 5^{x-2} = 600\sqrt{5}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4.5

Answer

The value of

x

is 4.5.

Factoring out

5^{x-2}

converts the left side into

5^{x-2}(25 - 1) = 24 \cdot 5^{x-2}

. Dividing

600\sqrt{5}

by 24 gives

25\sqrt{5}

, which equals

5^{2.5}

. Setting

x - 2 = 2.5

gives

x = 4.5

.

Step-by-Step Solution

1

Factor out the smallest power of 5, which is

5^{x-2}

, from the left side of the equation.

5^{x-2}(5^2 - 1) = 600\sqrt{5}

Factoring isolates the constant multiplier from the variable power term.

2

Calculate the numerical value inside the parentheses.

24 \cdot 5^{x-2} = 600\sqrt{5}

5^2 - 1 = 25 - 1 = 24

.

3

Divide both sides of the equation by 24.

5^{x-2} = 25\sqrt{5}

Isolating

5^{x-2}

gives

\frac{600\sqrt{5}}{24} = 25\sqrt{5}

.

4

Rewrite

25\sqrt{5}

as a single exponential expression with base 5.

5^{x-2} = 5^2 \cdot 5^{0.5} = 5^{2.5}

Using the product rule for exponents,

5^2 \cdot 5^{1/2} = 5^{2 + 0.5} = 5^{2.5}

.

5

Equate the exponents since the bases are identical.

x - 2 = 2.5 \implies x = 4.5

For any positive base

b \neq 1

, if

b^m = b^n

, then

m = n

.

Key Concept

Exponents, Roots, and Powers of Integers

Question 8Question →

Let $n = 2^a \cdot 3^b \cdot 5$ , where $a$ and $b$ are positive integers. If $n$ has a total of 24 positive divisors and has an equal number of even positive divisors and odd positive divisors, what is the value of $b$ ?

Show answer & explanation

Answer: 5

Answer

5

For

n = 2^a \cdot 3^b \cdot 5^1

, the total number of divisors is

(a + 1)(b + 1)(2) = 24

, giving

(a + 1)(b + 1) = 12

. The odd divisors are formed strictly from

3^b \cdot 5^1

, giving

(b + 1)(2)

odd divisors. The even divisors require at least one factor of 2, giving

a(b + 1)(2)

even divisors. Setting even and odd divisor counts equal gives

2a(b + 1) = 2(b + 1)

, which reduces to

a = 1

. Substituting

a = 1

into

(a + 1)(b + 1) = 12

gives

2(b + 1) = 12

, leading directly to

b = 5

.

Step-by-Step Solution

1

Set up the formula for total positive divisors of

n

.

(a + 1)(b + 1)(1 + 1) = 24 \implies (a + 1)(b + 1) = 12

The total number of divisors of

p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}

is

(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)

.

2

Determine the counts of odd and even positive divisors.

Odd divisors =

2(b + 1)

, Even divisors =

2a(b + 1)

Odd divisors cannot contain any factors of 2 (exponent of 2 is 0). Even divisors must contain at least one factor of 2 (exponent of 2 can be

1, 2, \dots, a

).

3

Equate the number of odd and even divisors to find

a

.

2a(b + 1) = 2(b + 1) \implies a = 1

Dividing both sides by

2(b + 1)

(which is positive since

b \ge 1

) leaves

a = 1

.

4

Solve for

b

using the total divisor relation.

(1 + 1)(b + 1) = 12 \implies 2(b + 1) = 12 \implies b = 5

Substituting

a = 1

into

(a + 1)(b + 1) = 12

yields

b = 5

.

Key Concept

Counting total, odd, and even positive divisors using prime factorization exponents

Question 9Question →

If $n$ is a positive integer such that $\frac{6^{n+3} \cdot 15^{2n-1}}{10^{n+1} \cdot 9^{n+1}} = 8{}100$ , what is the value of $n$ ?

Show answer & explanation

Answer: 4

Answer

4

Rewriting all composite bases (

6, 15, 10, 9

) into prime bases (

2, 3, 5

) simplifies the equation to

4 \cdot 3^n \cdot 5^{n-2} = 8,100

. Dividing by 4 yields

3^n \cdot 5^{n-2} = 2,025 = 3^4 \cdot 5^2

, which gives

n = 4

.

Step-by-Step Solution

1

Express each composite base in the fraction using its prime factor decomposition.

Numerator:

(2 \cdot 3)^{n+3} \cdot (3 \cdot 5)^{2n-1} = 2^{n+3} \cdot 3^{n+3} \cdot 3^{2n-1} \cdot 5^{2n-1} = 2^{n+3} \cdot 3^{3n+2} \cdot 5^{2n-1}

.
Denominator:

(2 \cdot 5)^{n+1} \cdot (3^2)^{n+1} = 2^{n+1} \cdot 5^{n+1} \cdot 3^{2n+2}

.

Breaking composite numbers down into prime factors allows terms with identical bases to be combined using exponent rules.

2

Simplify the fraction by subtracting the denominator exponents from the numerator exponents for each prime base.

\frac{2^{n+3} \cdot 3^{3n+2} \cdot 5^{2n-1}}{2^{n+1} \cdot 3^{2n+2} \cdot 5^{n+1}} = 2^{(n+3)-(n+1)} \cdot 3^{(3n+2)-(2n+2)} \cdot 5^{(2n-1)-(n+1)} = 2^2 \cdot 3^n \cdot 5^{n-2} = 4 \cdot 3^n \cdot 5^{n-2}

.

Applying the quotient rule for exponents:

\frac{a^x}{a^y} = a^{x-y}

.

3

Set the simplified expression equal to

8,100

and solve for

n

.

4 \cdot 3^n \cdot 5^{n-2} = 8,100 \implies 3^n \cdot 5^{n-2} = 2,025

. Prime factorization of

2,025 = 81 \cdot 25 = 3^4 \cdot 5^2

. Equating exponents gives

n = 4

(and

n-2 = 2

, which confirms

n=4

).

Because 3 and 5 are distinct prime bases, the unique prime factorization theorem guarantees that

3^n \cdot 5^{n-2} = 3^4 \cdot 5^2

implies

n = 4

.

Key Concept

Simplifying exponential expressions using prime factorization and quotient laws

Estimated Time:2m 0s

Question 10Question →

For any positive integer $n$ , let $f(n)$ denote the product of all distinct prime factors of $n$ . For example, $f(12) = 2 \times 3 = 6$ . What is the value of $f(2^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4)$ ?

Show answer & explanation

Answer: 330

Answer

330

To find

f(2^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4)

, first factor out the greatest common factor

2^4 \cdot 3^2 \cdot 5^2

. This yields

2^4 \cdot 3^2 \cdot 5^2 (2^2 \cdot 3^2 - 5^2) = 2^4 \cdot 3^2 \cdot 5^2 (36 - 25) = 2^4 \cdot 3^2 \cdot 5^2 \cdot 11

. The distinct prime factors present in this expression are 2, 3, 5, and 11. Multiplying these distinct prime factors gives

2 \times 3 \times 5 \times 11 = 330

.

Step-by-Step Solution

1

Factor out the greatest common term

2^4 \cdot 3^2 \cdot 5^2

from

2^6 \cdot 3^4 \cdot 5^2 - 2^4 \cdot 3^2 \cdot 5^4

2^4 \cdot 3^2 \cdot 5^2 \cdot (2^2 \cdot 3^2 - 5^2)

Factoring out common prime powers simplifies the expression and avoids large calculations.

2

Evaluate the arithmetic expression inside the parentheses

2^2 \cdot 3^2 - 5^2 = 4 \cdot 9 - 25 = 36 - 25 = 11

11 is itself a prime number.

3

Write the full prime factorization of the overall number

2^4 \cdot 3^2 \cdot 5^2 \cdot 11^1

All bases (2, 3, 5, 11) are prime numbers, giving the complete prime factorization.

4

Multiply each distinct prime factor together to evaluate

f(n)

2 \times 3 \times 5 \times 11 = 330

The function

f(n)

takes the product of each unique prime factor exactly once.

Key Concept

Prime Factorization and Product of Distinct Prime Factors

Question 11Question →

A set $S$ consists of $k$ consecutive integers. The sum of the first $m$ integers in set $S$ is $34$ , and the sum of the last $m$ integers in set $S$ is $74$ . If the sum of all $k$ integers in set $S$ is $189$ , what is the value of $k$ ?

Show answer & explanation

Answer: 14

Answer

The total number of integers in set S is 14.

By applying the property that the arithmetic mean of an evenly spaced set is the average of the mean of its first

m

elements and the mean of its last

m

elements, we establish that the mean of the set is

\frac{54}{m}

. Since the total sum is

189

,

k \cdot \frac{54}{m} = 189

, which yields

k = 3.5m

. Substituting this ratio into the difference between the two subset sums

m(k-m) = 40

gives

2.5m^2 = 40

, so

m = 4

and

k = 14

.

Step-by-Step Solution

1

Set up algebraic expressions for the sums of the first m terms and last m terms.

Let the set be

S = \{a, a+1, \dots, a+k-1\}

. The first

m

terms sum to

S_1 = m a + \frac{m(m-1)}{2} = 34

. The last

m

terms sum to

S_2 = m(a+k-m) + \frac{m(m-1)}{2} = 74

.

Consecutive integer sums can be represented by the starting term and the number of terms.

2

Subtract the sum of the first m terms from the sum of the last m terms.

S_2 - S_1 = m(a+k-m) - ma = m(k-m) = 74 - 34 = 40

.

Subtracting eliminates the initial term

a

and quadratic term

\frac{m(m-1)}{2}

, giving a clean relationship between

m

and

k

.

3

Determine the arithmetic mean of set S using subset averages.

The average of the first

m

terms is

\frac{34}{m}

and the average of the last

m

terms is

\frac{74}{m}

. The average of the entire set is the midpoint of these two averages:

\text{Mean} = \frac{1}{2}\left(\frac{34}{m} + \frac{74}{m}\right) = \frac{54}{m}

.

In any evenly spaced set, the overall median/mean is equal to the average of the lower-bound subset mean and upper-bound subset mean.

4

Relate the total sum to the set size k and overall mean.

S_{total} = k \times \text{Mean} \implies 189 = k \left(\frac{54}{m}\right) \implies \frac{k}{m} = \frac{189}{54} = 3.5 \implies k = 3.5m

.

The sum of a set of consecutive integers is always equal to the number of terms times the mean of the set.

5

Solve for m and k using the system of equations.

Substitute

k = 3.5m

into

m(k-m) = 40 \implies m(2.5m) = 40 \implies 2.5m^2 = 40 \implies m^2 = 16 \implies m = 4

. Thus,

k = 3.5 \times 4 = 14

.

Since

m > 0

, taking the positive square root gives

m = 4

, which leads directly to

k = 14

.

Key Concept

Average and Sum Equivalences in Evenly Spaced Sets

Question 12Question →

How many integer values of $y$ satisfy the inequality $|2y - 7| \le |y + 2|$ ?

Show answer & explanation

Answer: 8

Answer

The total number of integer values of

y

satisfying the inequality is 8.

Squaring both sides of

|2y - 7| \le |y + 2|

gives

3y^2 - 32y + 45 \le 0

, which factors into

(3y - 5)(y - 9) \le 0

. The solution range for

y

is

\frac{5}{3} \le y \le 9

. Since

\frac{5}{3} \approx 1.67

, the integer values of

y

satisfying this range are

2, 3, 4, 5, 6, 7, 8, 9

, giving a total of 8 integers.

Step-by-Step Solution

1

Square both sides of the inequality

|2y - 7| \le |y + 2|

(2y - 7)^2 \le (y + 2)^2

Since both sides of an absolute value expression are non-negative, squaring both sides maintains the inequality direction.

2

Expand terms and move all terms to the left side

3y^2 - 32y + 45 \le 0

Expanding gives

4y^2 - 28y + 49 \le y^2 + 4y + 4

. Subtracting

(y^2 + 4y + 4)

from both sides produces the standard quadratic inequality.

3

Factor the quadratic expression to find critical points

(3y - 5)(y - 9) \le 0

, yielding

\frac{5}{3} \le y \le 9

The roots are

y = \frac{5}{3}

and

y = 9

. A quadratic with a positive leading coefficient is non-positive between its roots.

4

Determine all integers within the range

\left[\frac{5}{3}, 9\right]

2, 3, 4, 5, 6, 7, 8, 9

(8 integers total)

Because

\frac{5}{3} \approx 1.67

, the smallest integer within the range is 2 and the largest is 9.

Key Concept

Solving absolute value inequalities of the form

|A| \le |B|

by squaring both sides and determining integer solutions.

Question 13Question →

An investor allocated a total principal of $\$25,000$ between two savings accounts, Account X and Account Y. Account X earns simple interest at an annual rate of $8\%$ , while Account Y earns interest at an annual rate of $10\%$ , compounded annually. If after $2$ years the total interest earned from Account Y exceeds the total interest earned from Account X by $\$1,550$ , what was the amount, in dollars, invested in Account Y?

Show answer & explanation

Answer: 15000

Answer

The amount invested in Account Y was $15,000 dollars.

Let

P_Y

be the principal invested in Account Y. Since the total investment is

\$25,000

, the principal invested in Account X is

25,000 - P_Y

. Account X earns simple interest over 2 years equal to

(25,000 - P_Y) \times 0.08 \times 2 = 4,000 - 0.16 P_Y

. Account Y earns compound interest over 2 years equal to

P_Y \times ((1 + 0.10)^2 - 1) = 0.21 P_Y

. The problem states that the interest from Account Y exceeds that from Account X by

\$1,550

, giving the equation

0.21 P_Y - (4,000 - 0.16 P_Y) = 1,550

. Combining like terms yields

0.37 P_Y = 5,550

, which simplifies to

P_Y = 15,000

.

Step-by-Step Solution

1

Express the principal of Account X in terms of Account Y

Principal of Account X =

25,000 - P_Y

The total sum invested across both accounts is

\$25,000

.

2

Calculate the interest earned from Account X over 2 years

I_X = 4,000 - 0.16 P_Y

Simple interest is calculated as

I = P \times r \times t = (25,000 - P_Y) \times 0.08 \times 2

.

3

Calculate the compound interest earned from Account Y over 2 years

I_Y = 0.21 P_Y

The compound multiplier for 2 years at

10\%

is

(1.10)^2 = 1.21

, giving a interest percentage of

1.21 - 1 = 0.21

or

21\%

.

4

Set up and simplify the equation for the difference in interest

0.37 P_Y = 5,550

Subtracting

I_X

from

I_Y

yields

0.21 P_Y - (4,000 - 0.16 P_Y) = 1,550

, which simplifies to

0.37 P_Y - 4,000 = 1,550

.

5

Solve for

P_Y

P_Y = 15,000

Dividing

5,550

by

0.37

gives

15,000

.

Key Concept

Combining Simple and Compound Interest Linear Equations

Question 14Question →

An analyst at an investment firm evaluates a portfolio of $160$ corporate bonds for two potential risk factors: credit rating downgrade risk and liquidity risk. The evaluation reveals that $60$ bonds have credit rating downgrade risk, $72$ bonds have liquidity risk, and $64$ bonds have neither risk factor. If a bond is selected at random from those in the portfolio that have at least one of the two risk factors, what is the probability that it has credit rating downgrade risk? Express your answer as a decimal.

Show answer & explanation

Answer: 0.625

Answer

The probability is 0.625 (or 5/8).

To calculate the probability that a bond has credit rating downgrade risk given that it has at least one risk factor, the sample space must be restricted to bonds with at least one risk factor. Out of

160

bonds,

64

have neither risk factor, leaving

160 - 64 = 96

bonds with at least one risk factor. All

60

bonds with credit rating downgrade risk are part of this group. The required conditional probability is

\frac{60}{96} = \frac{5}{8} = 0.625

.

Step-by-Step Solution

1

Calculate the size of the restricted sample space (bonds with at least one risk factor).

Total bonds with at least one risk factor = 160 - 64 = 96 bonds.

The condition specifies that the selection is made only from bonds having at least one risk factor.

2

Identify the number of favorable outcomes within this restricted sample space.

Number of bonds with credit rating downgrade risk = 60.

All 60 bonds with credit rating downgrade risk inherently possess at least one risk factor, so they lie entirely within the restricted sample space.

3

Compute the conditional probability P(Downgrade Risk | At Least One Risk).

60 / 96 = 5 / 8 = 0.625.

Conditional probability requires dividing the count of favorable outcomes by the count of the restricted sample space.

Key Concept

Conditional Probability and Sample Space Restriction

Question 15Question →

How many integer values of $x$ satisfy the inequality $||2x - 5| - 7| \leq 4$ ?

Show answer & explanation

Answer: 10

Answer

The total number of integer values of

x

satisfying the inequality is 10.

To solve

||2x - 5| - 7| \leq 4

, rewrite the inequality without the outer absolute value as

-4 \leq |2x - 5| - 7 \leq 4

. Adding 7 across all parts yields

3 \leq |2x - 5| \leq 11

. The inequality

|2x - 5| \leq 11

simplifies to

-3 \leq x \leq 8

. The inequality

|2x - 5| \geq 3

simplifies to

x \leq 1

or

x \geq 4

. Intersecting these two regions gives the set of real numbers

x \in [-3, 1] \cup [4, 8]

. The integer solutions within

[-3, 1]

are

-3, -2, -1, 0, 1

(5 integers), and within

[4, 8]

are

4, 5, 6, 7, 8

(5 integers). The total number of valid integer solutions is

5 + 5 = 10

.

Step-by-Step Solution

1

Unfold the outer absolute value expression.

-4 \leq |2x - 5| - 7 \leq 4

An inequality of the form

|U| \leq C

with

C > 0

is equivalent to

-C \leq U \leq C

.

2

Isolate the inner absolute value expression by adding 7 throughout.

3 \leq |2x - 5| \leq 11

Adding a constant to all parts preserves the direction of the inequality.

3

Solve the upper bound

|2x - 5| \leq 11

.

-3 \leq x \leq 8

-11 \leq 2x - 5 \leq 11

adds 5 to give

-6 \leq 2x \leq 16

, which divides by 2 to yield

-3 \leq x \leq 8

.

4

Solve the lower bound

|2x - 5| \geq 3

.

x \leq 1

or

x \geq 4

An inequality

|U| \geq C

splits into

U \geq C

(

2x - 5 \geq 3 \implies x \geq 4

) or

U \leq -C

(

2x - 5 \leq -3 \implies x \leq 1

).

5

Find the intersection of the upper and lower bound conditions and count the integers.

10 integer solutions:

\{-3, -2, -1, 0, 1, 4, 5, 6, 7, 8\}

.

Combining

-3 \leq x \leq 8

with (

x \leq 1

or

x \geq 4

) produces two disjoint intervals

[-3, 1]

and

[4, 8]

, containing 5 integers each.

Key Concept

Solving nested absolute value inequalities using double inequalities and boundary region intersections.

Question 16Question →

What is the sum of all integer values of $x$ that satisfy the inequality $|x^2 - 6x + 5| < 2x - 2$ ?

Show answer & explanation

Answer: 15

Answer

The sum of all integer values of

x

satisfying the inequality is

15

.

Factoring both sides gives

|(x - 1)(x - 5)| < 2(x - 1)

. Since the absolute value is non-negative, the right-hand side requires

2x - 2 > 0 \implies x > 1

. Under

x > 1

, the term

x - 1

is strictly positive, allowing us to simplify to

|x - 5| < 2

, which yields

3 < x < 7

. The integer solutions are

4, 5,

and

6

, and their sum is

15

.

Step-by-Step Solution

1

Determine the necessary condition for the right-hand side of the inequality.

Because the left-hand side

|x^2 - 6x + 5|

is non-negative for all real

x

, the right-hand side must be strictly positive. Thus,

2x - 2 > 0 \implies x > 1

.

An absolute value expression cannot be strictly less than a zero or negative quantity.

2

Factor the quadratic expression inside the absolute value and the linear expression on the right.

|(x - 1)(x - 5)| < 2(x - 1)

.

Factoring reveals a common linear factor

(x - 1)

on both sides.

3

Simplify the inequality using the condition

x > 1

.

Since

x > 1

, we know

x - 1 > 0

, so

|x - 1| = x - 1

. Splitting the product inside the absolute value gives

(x - 1)|x - 5| < 2(x - 1)

. Dividing both sides by

(x - 1)

yields

|x - 5| < 2

.

Dividing an inequality by a strictly positive number preserves the direction of the inequality sign.

4

Solve the simplified absolute value inequality and sum the integer solutions.

|x - 5| < 2 \implies -2 < x - 5 < 2 \implies 3 < x < 7

. The integer solutions strictly within this range are

x = 4, 5,

and

6

. Their sum is

4 + 5 + 6 = 15

.

The integers strictly between

3

and

7

are

4

,

5

, and

6

.

Key Concept

Solving Quadratic Absolute Value Inequalities via Domain Constraints and Factoring

Estimated Time:2m 0s

Question 17Question →

A manufacturing shop produces two models of custom bicycle frames: Standard and Deluxe. Producing each Standard frame requires $2$ hours of welding and $1$ hour of painting. Producing each Deluxe frame requires $3$ hours of welding and $2$ hours of painting. During a single week, the shop logged a total of $130$ hours of welding and $75$ hours of painting for these two models. How many Deluxe bicycle frames were produced during that week?

Show answer & explanation

Answer: 20

Answer

The number of Deluxe bicycle frames produced during that week is 20.

Translating the resource limitations into a system of two linear equations yields

2x + 3y = 130

for welding hours and

x + 2y = 75

for painting hours, where

x

and

y

represent the number of Standard and Deluxe frames respectively. Expressing

x

in terms of

y

from the painting equation gives

x = 75 - 2y

. Substituting this expression into the welding equation gives

2(75 - 2y) + 3y = 130

, which simplifies to

150 - y = 130

, giving

y = 20

Deluxe frames.

Step-by-Step Solution

1

Define variables for the unknowns.

Let

x

be the number of Standard bicycle frames produced and

y

be the number of Deluxe bicycle frames produced.

Assigning variables allows us to translate the problem into algebraic expressions.

2

Set up a system of linear equations.

Welding constraint:

2x + 3y = 130

Painting constraint:

x + 2y = 75

Each constraint represents the sum of hours spent on Standard and Deluxe frames for that process.

3

Solve the system using elimination or substitution.

Multiply the painting equation by

2

:

2x + 4y = 150

.
Subtract the welding equation (

2x + 3y = 130

) from this result:

(2x + 4y) - (2x + 3y) = 150 - 130 \Rightarrow y = 20

.

Eliminating

x

directly solves for

y

, which is the requested quantity (Deluxe frames).

Key Concept

Solving systems of two linear equations with two variables by substitution or elimination.

Estimated Time:1m 30s

Question 18Question →

A company recorded the number of daily sales over a 7-day week. The numbers of daily sales, when arranged in ascending order, are $12, 15, 18, x, 22, y, 30$ . If the median of the 7 daily sales figures is $20$ and the arithmetic mean is $21$ , what is the value of $y$ ?

Show answer & explanation

Answer: 30

Answer

The value of

y

is 30.

For an ordered set of 7 numbers, the median is the 4th term, which means

x = 20

. The total sum of the set is found by multiplying the number of terms by the mean:

7 \times 21 = 147

. Adding all known terms (

12 + 15 + 18 + 20 + 22 + 30

) gives 117. Subtracting 117 from 147 yields

y = 30

.

Step-by-Step Solution

1

Determine the value of

x

using the median definition.

x = 20

Since the 7 numbers are given in ascending order, the median is the middle (4th) term.

2

Calculate the total sum of all 7 numbers.

Total sum =

147

The sum of elements in a set equals the number of elements multiplied by the arithmetic mean (

7 \times 21 = 147

).

3

Sum all known numbers and set up an equation for

y

.

117 + y = 147

12 + 15 + 18 + 20 + 22 + 30 = 117

.

4

Solve for

y

.

y = 30

Subtract 117 from 147 to isolate

y

.

Key Concept

Using properties of median and arithmetic mean to find missing terms in an ordered data set.

Question 19Question →

At an agricultural cooperative, $0.45$ of the total fruit harvested by weight consisted of apples, and $\frac{3}{8}$ of the remaining fruit weight consisted of pears. If the remaining $550$ kilograms of fruit consisted entirely of oranges, what was the total weight, in kilograms, of the fruit harvested?

Show answer & explanation

Answer: 1600

Answer

1600

Subtracting the apple portion leaves 55% (or 11/20) of the total harvest. Since pears make up 3/8 of this remainder, oranges make up the remaining 5/8 of the 11/20 portion, which simplifies to 11/32 of the total harvest. Setting 11/32 of the total harvest equal to 550 kg yields a total weight of 1,600 kg.

Step-by-Step Solution

1

Find the fraction of the total harvest left after subtracting the apples.

The remaining portion is 1 - 0.45 = 0.55, which equals 11/20 of the total harvest.

Apples account for 0.45 of the harvest.

2

Calculate the portion of the harvest represented by oranges.

Since pears are 3/8 of the remainder, oranges are 5/8 of the remainder. Thus, oranges are (5/8) * (11/20) = 11/32 of the total harvest.

The non-apple harvest consists only of pears and oranges.

3

Solve for the total weight using the given weight of oranges.

Total weight = 550 * (32 / 11) = 1600 kg.

11/32 of the total weight equals 550 kilograms.

Key Concept

Multi-step arithmetic combining decimals and fractions to solve remaining-quantity word problems

Question 20Question →

What is the exponent of $3$ in the prime factorization of the integer $S = 25! + 26! + 27!$ ?

Show answer & explanation

Answer: 16

Answer

The exponent of

3

in the prime factorization of

S

is

16

.

Factoring

25!

from the sum gives

S = 25!(1 + 26 + 26 \times 27) = 25!(729) = 25! \times 3^6

. Applying Legendre's formula to

25!

yields

\lfloor 25/3 \rfloor + \lfloor 25/9 \rfloor = 8 + 2 = 10

factors of

3

. Adding the

6

factors of

3

from

729 = 3^6

gives a total exponent of

10 + 6 = 16

.

Step-by-Step Solution

1

Factor out

25!

from the sum

S = 25! + 26! + 27!

S = 25! \left(1 + 26 + 26 \times 27\right)

Factoring out the greatest common factorial term

25!

converts the sum into a product of

25!

and an integer factor.

2

Simplify the expression inside the parentheses

1 + 26 + 702 = 729 = 3^6

Using algebraic simplification,

1 + 26(1 + 27) = 1 + 26(28) = 729 = 27^2 = (3^3)^2 = 3^6

.

3

Compute the exponent of

3

in

25!

using Legendre's formula

E_3(25!) = \lfloor \frac{25}{3} \rfloor + \lfloor \frac{25}{9} \rfloor + \lfloor \frac{25}{27} \rfloor = 8 + 2 + 0 = 10

Legendre's formula counts the total prime factors of

3

contributed by all multiples of

3, 9, 27, \dots

up to

25

.

4

Combine the exponent of

3

from

25!

and the factor

729

Total exponent of

3 = 10 + 6 = 16

Since

S = 25! \times 3^6 = (3^{10} \times k) \times 3^6 = 3^{16} \times k

(where

3 \nmid k

), the total exponent of

3

is

10 + 6 = 16

.

Key Concept

Exponent of a prime in factorial expressions using Legendre's formula and algebraic factoring

Answer

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