Question
In the number \(12ab\), \(a\) and \(b\) are digits. Find \(a\) and \(b\) such that the number \(12ab\) is divisible by 2, 5 and 7.
Solution
For the number \(12ab\) to be divisible by 2, \(b\) must be 0, 2, 4, 6 or 8. For the same number to be divisible by 5, \(b\) must be 0 or 5. For \(12ab\) to be divisible by 2 and 5, \(b\) must be equal to 0. Hence \(b = 0\).
We now write the number as:
\[12ab = 1200 + 10a + b\]The division of 1200 by 7 gives a quotient of 171 and a remainder of 3 and \(b = 0\). Hence:
\[12ab = 7 \times 171 + 3 + 10a + 0\]For \(12ab\) to be divisible by 7, \(10a + 3\) must be divisible by 7. Testing values:
- \(a = 0: 10(0) + 3 = 3\) (not divisible by 7)
- \(a = 1: 10(1) + 3 = 13\) (not divisible by 7)
- \(a = 2: 10(2) + 3 = 23\) (not divisible by 7)
- \(a = 3: 10(3) + 3 = 33\) (not divisible by 7)
- \(a = 4: 10(4) + 3 = 43\) (not divisible by 7)
- \(a = 5: 10(5) + 3 = 53\) (not divisible by 7)
- \(a = 6: 10(6) + 3 = 63\) (divisible by 7)
Thus \(a = 6\) and \(b = 0\).
Question
What is the average (arithmetic mean) of all multiples of 6 from 6 to 510 inclusive?
Solution
The average \(A\) of all multiples of 6 from 6 to 510 is:
\[A = \frac{S}{n}\]where \(S = 6 + 12 + ... + 510\) and \(n\) is the number of multiples.
Since \(S = \frac{n}{2}(6 + 510)\), then:
\[A = \frac{1}{2}(6 + 510) = 258\]Question
Which of the following numbers can be expressed as the product of 3 different integers greater than 1?
I) 24
II) 27
III) 48
A) I and III only
B) I and II only
C) I, II and III
D) I only
E) III only
Solution
Prime factorizations:
- \(24 = 2^3 \times 3 = 2 \times 4 \times 3\) ✓
- \(27 = 3^3 = 3 \times 9\) (only two different integers) ✗
- \(48 = 2^4 \times 3 = 2 \times 8 \times 3\) ✓
Answer: A) I and III only.
Question
If 60% of \(a\) is equal to 120% of \(b\), then \(2a - 4b = ?\)
Solution
Translate the statement:
\[0.6a = 1.2b\]Simplify:
\[6a = 12b\] \[a = 2b\]Then:
\[2a - 4b = 2(2b) - 4b = 4b - 4b = 0\]Question
\[ \frac{4^{1000} \times 8^{2000}}{16^{2000}} = ? \]
Solution
Express as powers of 2:
\(4 = 2^2\), \(8 = 2^3\), \(16 = 2^4\)
\[ \frac{(2^2)^{1000} \times (2^3)^{2000}}{(2^4)^{2000}} = \frac{2^{2000} \times 2^{6000}}{2^{8000}} = \frac{2^{8000}}{2^{8000}} = 1 \]Question
If \(n\) is a positive integer divisible by 2, 5 and 7, which of the following is true?
I) \(n^2 + 100\) is divisible by 100
II) \(n + 28\) is divisible by 14
III) \(n^2 + 1\) is divisible by 2
A) I and II only
B) I, II and III
C) None
D) I and III only
E) II and III only
Solution
Since \(n\) is divisible by 2 and 5, \(n\) is a multiple of 10 ⇒ \(n^2\) is multiple of 100 ⇒ \(n^2 + 100\) divisible by 100. ✓
Since \(n\) is divisible by 2 and 7, \(n\) is multiple of 14 ⇒ \(n + 28\) divisible by 14. ✓
Since \(n\) is divisible by 2, \(n^2\) is even ⇒ \(n^2 + 1\) is odd ⇒ not divisible by 2. ✗
Answer: A) I and II only.
Question
A spherical block of metal weighs 12 pounds. What is the weight, in pounds, of another block of the same metal if its radius is 3 times the radius of the 12-pound block?
Solution
Volume of sphere: \(V = \frac{4}{3}\pi r^3\)
Weight is proportional to volume. If radius triples:
\[ \frac{V_2}{V_1} = \left(\frac{3r}{r}\right)^3 = 27 \]New weight: \(27 \times 12 = 324\) pounds.
Question
If the ages of students in a class vary between 18 and 22 inclusive, which of the following describes all possible ages \(x\) in this class?
A) \(|x - 18| \le 4\)
B) \(|x + 20| \le 2\)
C) \(|x - 21| \le 1\)
D) \(|x - 22| \le 0\)
E) \(|x - 20| \le 2\)
Solution
The condition: \(18 \le x \le 22\)
Subtract 20: \(-2 \le x - 20 \le 2\)
Which is equivalent to: \(|x - 20| \le 2\)
Answer: E.
Question
If \(r\%\) of \(y\) is \(A\), what is \(y\)?
Solution
Translate:
\[ \frac{r}{100} \times y = A \] \[ y = \frac{100A}{r} \]Question
Five years ago Ben was 4 times as old as Julie. If Julie is 10 years old now, how old is Ben?
Solution
Let Ben's current age = \(x\).
Five years ago: Ben = \(x - 5\), Julie = \(10 - 5 = 5\)
Equation: \(x - 5 = 4 \times 5\)
\[ x - 5 = 20 \] \[ x = 25 \]Question
If \(\frac{x + 2}{7}\) is an integer greater than 2, then the remainder when \(x\) is divided by 7 is:
Solution
Let \(\frac{x + 2}{7} = k\) where \(k > 2\) is an integer.
\[ x + 2 = 7k \] \[ x = 7k - 2 = 7(k - 1) + 5 \]Thus remainder is 5.
Question
For how many positive integer values of \(n\) will \(2n^2 + 1\) be an integer greater than 2 and smaller than 400?
Solution
Solve inequality:
\[ 2 < 2n^2 + 1 < 400 \] \[ 1 < 2n^2 < 399 \] \[ \frac{1}{2} < n^2 < \frac{399}{2} \] \[ \sqrt{\frac{1}{2}} < n < \sqrt{\frac{399}{2}} \]Since \(n\) positive integer: \(1 \le n \le 14\)
Number of integers: 14.
Question
If \(x + y = \sqrt{22}\) and \(x - y = \sqrt{10}\), then \(xy = ?\)
Solution
Square both equations:
\[ (x + y)^2 = 22 \Rightarrow x^2 + 2xy + y^2 = 22 \] \[ (x - y)^2 = 10 \Rightarrow x^2 - 2xy + y^2 = 10 \]Subtract second from first:
\[ (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 22 - 10 \] \[ 4xy = 12 \] \[ xy = 3 \]Question
In the figure below, the 3 circles have equal radii and are tangent to each other and to the sides of the rectangle. The width of the rectangle is 20 feet long. What is the area of the shaded region?
Solution
Let radius = \(r\). From width: \(4r = 20 \Rightarrow r = 5\).
The centers form equilateral triangle with side \(2r = 10\).
Area of triangle: \(A_t = \frac{1}{2} \times 10 \times 10 \times \sin 60^\circ = 25\sqrt{3}\)
Area of one sector (60°): \(A_s = \frac{60}{360} \times \pi r^2 = \frac{1}{6} \times \pi \times 25 = \frac{25\pi}{6}\)
Shaded area = triangle area − 3 sector areas:
\[ 25\sqrt{3} - 3 \times \frac{25\pi}{6} = 25\sqrt{3} - \frac{25\pi}{2} = \frac{25(2\sqrt{3} - \pi)}{2} \]Question
If \(2x + 1 = -x + 3\), what is the value of \(\frac{2}{3}x + \frac{1}{3}\)?
Solution
Solve for \(x\):
\[ 2x + 1 = -x + 3 \] \[ 3x = 2 \] \[ x = \frac{2}{3} \]Substitute:
\[ \frac{2}{3}x + \frac{1}{3} = \frac{2}{3} \times \frac{2}{3} + \frac{1}{3} = \frac{4}{9} + \frac{3}{9} = \frac{7}{9} \]Question
If \(a < b < c < d\) and the average of \(a, b, c, d\) is \(m\), which of the following is true?
I) \(a + b < c + d\)
II) \(m < d\) and \(a < m\)
III) \(b < m < c\)
A) II only
B) I only
C) III only
D) I and II only
E) II and III only
Solution
I) Since \(a < c\) and \(b < d\), adding: \(a + b < c + d\). ✓
II) \(m = \frac{a+b+c+d}{4}\). Since \(a\) is smallest, \(a < m\). Since \(d\) is largest, \(m < d\). ✓
III) Not always true. Counterexample: \(a=3, b=10, c=11, d=12\), then \(m=9\), but \(b=10 > m\). ✗
Answer: D) I and II only.
Question
If \(x, y, z\) are positive numbers such that \(5x = \frac{y}{4}\), \(\frac{y}{4} = \frac{z}{5}\) and \(y + z = \frac{x}{k}\), what is \(k\)?
Solution
From first: \(y = 20x\)
From second: \(z = \frac{5}{4}y = \frac{5}{4} \times 20x = 25x\)
Substitute into third:
\[ y + z = 20x + 25x = 45x = \frac{x}{k} \] \[ 45 = \frac{1}{k} \] \[ k = \frac{1}{45} \]Question
Two square tables have sides of 10 and 15 inches. The area of the larger table is what percent more than the area of the smaller table?
Solution
Areas: \(A_s = 10^2 = 100\), \(A_l = 15^2 = 225\)
Percent more:
\[ \frac{225 - 100}{100} \times 100\% = 125\% \]Question
Find \(-a + b - c + d\) if \((x - \frac{1}{5})(x + 1)(x + \frac{3}{7}) = ax^3 + bx^2 + cx + d\) for all real \(x\).
Solution
Set \(x = -1\):
Left side: \((-1 - \frac{1}{5})(-1 + 1)(-1 + \frac{3}{7}) = 0\)
Right side: \(a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d\)
Thus \(-a + b - c + d = 0\).
Question
In the figure below line \(l\) is parallel to line \(k\) and line \(g\) is parallel to line \(h\). What is the value of \(y\)?
A) 10
B) 20
C) 5
D) 15
E) 35
Question
Line \(l\) has equation \(y - 2x = 2\). What is the equation of line \(p\) which is the reflection of line \(l\) on the line \(y = x\)?
A) \(y = 0.5x - 1\)
B) \(y = 2x - 1\)
C) \(y = x + 2\)
D) \(y = -2x - 2\)
E) \(y = x\)
Question
Mike drove 30 miles at constant speed for \(t\) hours and then drove \(y\) miles at another constant speed for 1 hour 15 minutes. What was his average speed for the whole journey?
A) \(\frac{\frac{30}{t} + \frac{y}{1.25}}{2}\)
B) \(\frac{30 + y}{t + 1.25}\)
C) \(\frac{30 + y}{t}\)
D) \(\frac{30 + y}{1.25}\)
E) \(\frac{30 + y}{2(t + 1.25)}\)
Question
If \(M\) and \(N\) are negative integers and \(-3M + 4N = 10\), which could be \(N\)?
A) \(-2\)
B) \(-4\)
C) \(-6\)
D) \(-7\)
E) \(-9\)
Question
If \(x^6 = 20\), what is \(x^8\)?
A) \(400\sqrt[3]{20}\)
B) \(10\sqrt{20}\)
C) \(20\sqrt{20}\)
D) \(20\sqrt[3]{20}\)
E) \(\sqrt[3]{20}\)
Question
\(n\) is chosen randomly from \(\{2,5,6\}\) and \(p\) from \(\{6,9,10\}\). What is the probability that both \(n\) and \(p\) are even?
A) \(\frac{2}{9}\)
B) \(\frac{5}{9}\)
C) \(\frac{4}{9}\)
D) \(\frac{1}{9}\)
E) \(\frac{7}{9}\)
Question
Three collinear points A, B, C with B between A and C. The distance AB is 8 more than 4 times BC, and AC is 5 times BC. What is AC?
A) 4
B) 20
C) 16
D) 24
E) 12