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Given that the average of a set of 10 numbers is 85, what is the average of the remaining numbers if the numbers 70 and 76 are excluded from the set? The sum of the 10 numbers is $10 \times 85 = 850$.
If we exclude the numbers 70 and 76, we subtract their sum from the total sum, so we have $850 - (70 + 76) = 850 - 146 = 704$.
There are now 8 numbers remaining in the set.
To find their average, we divide the sum by the number of numbers: $\frac{704}{8} = \boxed{88}$.
The answer is: 88
Let $P(x) = 0$ be the polynomial equation of least possible degree, with rational coefficients, having $\sqrt[3]{7} + \sqrt[3]{49}$ as a root. The product of all of the roots of $P(x) = X$ is 56. What is the value of unknown variable X? To solve this problem, we need to determine the value of x, which represents the unknown variable.
We are given that $\sqrt[3]{7} + \sqrt[3]{49}$ is a root of the polynomial equation $P(x) = 0$.
Let's break down the information given:
Root of the polynomial equation: $\sqrt[3]{7} + \sqrt[3]{49}$
Product of all the roots: 56
We can set up the equation as follows:
$(x - (\sqrt[3]{7} + \sqrt[3]{49})) = 0$
Since the product of all the roots is 56, we can write:
$(x - (\sqrt[3]{7} + \sqrt[3]{49}))(x - r_2)(x - r_3)...(x - r_n) = 56$
where $r_2, r_3, ..., r_n$ are the other roots of the polynomial equation.
Since we are looking for the value of x, we can expand the equation and set it equal to 0:
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})x^2 + (\sqrt[3]{7} + \sqrt[3]{49})^2x - (\sqrt[3]{7} + \sqrt[3]{49})^3 = 0$
Now, we can simplify and solve for x:
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})x^2 + (\sqrt[3]{7} + \sqrt[3]{49})^2x - (\sqrt[3]{7} + \sqrt[3]{49})^3 = 0$
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})(x^2 - (\sqrt[3]{7} + \sqrt[3]{49})x + (\sqrt[3]{7} + \sqrt[3]{49})^2) = 0$
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})(x^2 - 2\sqrt[3]{7}\cdot x + 7 + 2\sqrt[3]{7}\cdot x + 49) = 0$
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})(x^2 + 56) = 0$
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})x^2 - 56(\sqrt[3]{7} + \sqrt[3]{49}) = 0$
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})x^2 - 56\sqrt[3]{7} - 56\sqrt[3]{49} = 0$
Since we are given that the polynomial equation has rational coefficients, we know that the irrational terms will cancel out.
Therefore, we can set up the equation as follows:
$x^3 - (\sqrt[3]{7} + \sqrt[3]{49})x^2 - 56\sqrt[3]{7} - 56\sqrt[3]{49} = 0$
$x^3 - x^2 - 56 = 0$
Now, we can solve this equation for x. However, we do not need to determine the exact value of x since the question only asks for the value of the unknown variable x.
Therefore, the value of x is unknown.
The answer is: 0
How many integers between 1 and 11 (inclusive) result in a repeating decimal when divided by 12? A repeating decimal occurs when the denominator has factors other than 2 and 5.
The prime factorization of 12 is $2^2 \cdot 3$.
We need to count the numbers between 1 and 11 that have factors of 2 or 3.
These numbers are: 2, 3, 4, 6, 8, 9, 10, 11.
There are $\boxed{8}$ integers between 1 and 11 (inclusive) that result in a repeating decimal when divided by 12.The answer is: 8
At the end of the twelfth day, how many pebbles will Murtha have collected if she starts with one pebble on the first day and collects one more pebble than the previous day on each subsequent day? On the first day, Murtha collects 1 pebble.
On the second day, she collects 1 + 1 = 2 pebbles.
On the third day, she collects 2 + 1 = 3 pebbles.
This pattern continues, with Murtha collecting 1 more pebble than the previous day, until the twelfth day.
On the twelfth day, Murtha collects 11 + 1 = 12 pebbles.
To find the total number of pebbles Murtha has collected after twelve days, we can add up the number of pebbles she collects each day.
This is equivalent to finding the sum of the first twelve positive integers, which is given by the formula $\frac{n(n+1)}{2}$, where n is the number of terms.
Using this formula with n = 12, we find that Murtha has collected $\frac{12(12+1)}{2} = \frac{12(13)}{2} = 6(13) = \boxed{78}$ pebbles at the end of the twelfth day.
The answer is: 78
June’s class has 25 students. 60% are girls and the rest are boys. 40% of the boys like playing basketball and the rest don't. The number of girls who like playing basketball is double the number of boys who don't like to. What percentage of the girls in the class like playing basketball?
|
Out of 25 students, 60% are girls, so there are 25 * 0.6 = 15 girls in the class.
The remaining students are boys, so there are 25 - 15 = 10 boys in the class.
40% of the boys like playing basketball, so there are 10 * 0.4 = 4 boys who like playing basketball.
The rest of the boys don't like playing basketball, so there are 10 - 4 = 6 boys who don't like playing basketball.
The number of girls who like playing basketball is double the number of boys who don't like to, so there are 6 * 2 = 12 girls who like playing basketball.
Therefore, the percentage of girls in the class who like playing basketball is (12 / 15) * 100 = 80%
#### 80
The answer is: 80
|
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Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120^\circ$ and $\stackrel \frown {BC} = 72^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$ is:
$\textbf{(A)}\ \frac{5}{18}\qquad \textbf{(B)}\ \frac{2}{9}\qquad \textbf{(C)}\ \frac{1}{4}\qquad \textbf{(D)}\ \frac{1}{3}\qquad \textbf{(E)}\ \frac{4}{9}$
e $\stackrel \frown {AB} = 120^\circ$, we have $\angle ACB = \frac{360^\circ - \stackrel \frown {AB} - \stackrel \frown {BC}}{2} = 84^\circ$. Therefore, $\angle ABC = 180^\circ - \angle BAC - \angle ACB = 180^\circ - x - 84^\circ = 96^\circ$.
Since $\triangle BAC$ is acute-angled, $x < 90^\circ$. Therefore, $\angle BAC + \angle ABC = x + 96^\circ < 180^\circ$, so $E$ lies inside $\triangle ABC$.
Since $OE$ is perpendicular to $AC$, $\angle OEB = 90^\circ$. Therefore, $\angle BEO = 180^\circ - \angle OBE - \angle OEB = 180^\circ - \angle OBE - 90^\circ = 90^\circ - \angle OBE$.
Since $\triangle BAC$ is inscribed in a circle with center $O$, $\angle BAC = 2\angle BOC$. Therefore, $\angle BOC = \frac{1}{2}\angle BAC = \frac{1}{2}x$.
Since $E$ lies on minor arc $AC$, $\angle AEC = \frac{1}{2}\stackrel \frown {AC} = \frac{1}{2}(360^\circ - \stackrel \frown {AB} - \stackrel \frown {BC}) = 78^\circ$. Therefore, $\angle BOC = \frac{1}{2}\angle AEC = \frac{1}{2}(78^\circ) = 39^\circ$.
Since $\angle BOC = 39^\circ$, $\angle BEO = 90^\circ - \angle OBE = 90^\circ - (90^\circ - \angle BOC) = \angle BOC = 39^\circ$.
Therefore, $\angle OBE = 90^\circ - \angle BEO = 90^\circ - 39^\circ = 51^\circ$.
The ratio of the magnitudes of $\angle OBE$ and $\angle BAC$ is $\frac{51^\circ}{x}$.
Since $x < 90^\circ$, $\frac{51^\circ}{x} > \frac{51^\circ}{90^\circ} = \frac{17}{30} = \frac{1}{2}\cdot \frac{34}{60} = \frac{1}{2}\cdot \frac{17}{30} = \frac{1}{2}\cdot \frac{1}{2}\cdot \frac{34}{60} = \frac{1}{4}\cdot \frac{34}{30} = \frac{1}{4}\cdot \frac{17}{15} = \frac{1}{4}\cdot \frac{1}{3} = \frac{1}{12}$.
Since $x < 90^\circ$, $\frac{1}{12} < \frac{1}{3}$.
Therefore, the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$ is $\boxed{\frac{1}{3}}$.The answer is: \frac{1}{3}
Determine the projection of the vector $\begin{pmatrix} 4 \\ 5 \end{pmatrix}$ onto the vector $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$. The projection of a vector $\mathbf{v}$ onto another vector $\mathbf{u}$ is given by the formula
\[\text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{\|\mathbf{u}\|^2} \mathbf{u},\]
where $\mathbf{v} \cdot \mathbf{u}$ is the dot product of $\mathbf{v}$ and $\mathbf{u}$,
and $\|\mathbf{u}\|$ is the magnitude of $\mathbf{u}$.
In this case, we have $\mathbf{v} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$ and $\mathbf{u} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$.
The dot product of $\mathbf{v}$ and $\mathbf{u}$ is $\mathbf{v} \cdot \mathbf{u} = (4)(2) + (5)(0) = 8$,
and the magnitude of $\mathbf{u}$ is $\|\mathbf{u}\| = \sqrt{2^2 + 0^2} = 2$.
Therefore, the projection of $\mathbf{v}$ onto $\mathbf{u}$ is
\[\text{proj}_{\mathbf{u}} \mathbf{v} = \frac{8}{2^2} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \boxed{\begin{pmatrix} 4 \\ 0 \end{pmatrix}}.\]
The answer is: \begin{pmatrix}4\0\end{pmatrix}
A department store offers a 10% discount for the amount exceeding $x of the customer's total charge. Jaco bought a pair of shoes for $74; 2 pairs of socks that cost $2 per pair; a bag that costs $42. How much will Jaco pay for those items?
If we know the answer to the above question is 118, what is the value of unknown variable x? Jaco bought a pair of shoes for $74. This amount does not exceed $x, so there is no discount for the shoes.
He bought 2 pairs of socks that cost $2 per pair, so the total cost of the socks is 2 * $2 = $4. Again, this amount does not exceed $x, so there is no discount for the socks.
He bought a bag that costs $42. This amount also does not exceed $x, so there is no discount for the bag.
The total amount Jaco will pay is the sum of the costs of the shoes, socks, and bag: $74 + $4 + $42 = $120.
We are given that the total amount Jaco will pay is $118, so we can write: $120 - 10% * (120 - x) = $118.
Simplifying the left side, we get: $120 - 0.1 * (120 - x) = $118.
Distributing the 0.1, we get: $120 - 12 + 0.1x = $118.
Combining like terms, we get: 0.1x = $10.
Dividing both sides by 0.1, we get: x = $100.
The value of x is $100.
#### 100
The answer is: 100
Alex has X pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?
If we know the answer to the above question is 100,000, what is the value of unknown variable X?
|
We want to find the value of $X$ in the given situation.
To solve this problem, we can use the concept of permutations.
Since each friend can receive more than one problem, we need to find the number of ways to distribute $X$ pieces of paper among 10 friends.
The number of ways to distribute $X$ pieces of paper among 10 friends can be calculated using the formula:
Number of ways = $10^X$
We are given that the number of ways is 100,000, so we can write:
$10^X = 100,000$
To find the value of $X$, we can take the logarithm of both sides of the equation:
$\log_{10}(10^X) = \log_{10}(100,000)$
Using the logarithmic property $\log_{a}(a^X) = X$, we can simplify the equation to:
$X = \log_{10}(100,000)$
Calculating the logarithm, we find:
$X = 5$
The value of X is 5.
The answer is: 5
|
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"Beth had x students in her 10th-grade class. The following year 30 more students join. In her fin(...TRUNCATED) | "The school uses 200 sheets of paper per class per day, so in one week, each class uses 200 sheets/d(...TRUNCATED) | [33,2788,550,2124,2444,287,607,838,400,12,9526,1398,13,220,383,1708,614,1542,517,2444,4654,13,220,55(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"If Adam's father deposited $2000 in the bank, with 8% interest paid annually and the interest being(...TRUNCATED) | "To solve this problem, we need to determine the value of x, which represents the width of the sidew(...TRUNCATED) | [1532,7244,338,2988,27163,720,11024,287,262,3331,11,351,807,4,1393,3432,13844,290,262,1393,852,23019(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"Nina wants to give presents to her children. She buys three toys at $10 each, two packs of basketba(...TRUNCATED) | "Let's assume the cost of the sandwich is x.\nThe juice is two times more expensive than the sandwic(...TRUNCATED) | [45,1437,3382,284,1577,10969,284,607,1751,13,1375,24779,1115,14958,379,720,940,1123,11,734,15186,286(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"In the diagram, the rectangle has a width $w$, a length of X$, and a perimeter of $24$. What is the(...TRUNCATED) | "The jar of grape candy had 5 more pieces of candy than the jar of banana candy, so it had 43 + 5 = (...TRUNCATED) | [818,262,16362,11,262,35991,468,257,9647,720,86,47113,257,4129,286,1395,47113,290,257,25317,286,720,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"How many two-digit positive integers are congruent to X (mod 3)?\nIf we know the answer to the abov(...TRUNCATED) | "The cost of the new computer is $600 and it lasts for 6 years, so the cost per year is $600 / 6 = $(...TRUNCATED) | [2437,867,734,12,27003,3967,37014,389,369,48929,298,284,1395,357,4666,513,19427,198,1532,356,760,262(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"After Daragh took out her favorite 8 bears from her initial collection of 20 stuffed bears and divi(...TRUNCATED) | "Jimmy makes 8 sandwiches in total, using x slices of bread each, so the total number of slices of b(...TRUNCATED) | [3260,7491,10471,1718,503,607,4004,807,13062,422,607,4238,4947,286,1160,22259,13062,290,9086,262,563(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"For a project, a builder purchased 7 boxes of bolts with each box containing 11 bolts. He purchase(...TRUNCATED) | "The increase in price is $102 - $85 = $17.\nTo find the percent increase, we divide the increase by(...TRUNCATED) | [1890,257,1628,11,257,27098,8155,767,10559,286,24667,351,1123,3091,7268,1367,24667,13,220,679,8155,2(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
"There are 28 garden gnomes in a yard. Three-fourths of them have red hats, and the rest have blue h(...TRUNCATED) | "Jason eats an omelet made using x eggs every morning.\nIn two weeks, there are 14 days.\nSo the tot(...TRUNCATED) | [1858,389,2579,11376,19967,2586,287,257,12699,13,7683,12,14337,9998,286,606,423,2266,23910,11,290,26(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) |
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