Set 8

Explanation

Let $x = 0.323323323...$. Since there are three repeating digits ($323$), multiply both sides by $1000$.

Next, subtract the original equation $x = 0.323323...$ from this equation:

Thus, we get:

So, the common fraction form of the number is $\frac{323}{999}$.

Explanation

The sequence above consists of two alternating sub-sequences (skipping one number).

Pattern 1 (Odd positions)

The differences between consecutive terms are:

The pattern is subtracting consecutive odd numbers ($-3, -5, -7$). Thus, the next difference is $-9$. Next term: $27 - 9 = 18$.

Pattern 2 (Even positions)

The differences between consecutive terms are:

The pattern is subtracting consecutive even numbers ($-4, -6, -8$). Thus, the next difference is $-10$. Next term: $22 - 10 = 12$.

So, the next two numbers are $18$ and $12$.

Explanation

Note that:

Then the value of $A$ is:

And the value of $B$ is:

Thus, $A > B$.

Explanation

The new data follows the pattern:

The new average can be obtained by:

First, let's find the sum of the arithmetic series $3, 5, 7, ...$. This sequence has the first term $a = 3$ and difference $b = 2$. The 10th term is:

The sum of the first 10 terms is:

The new average can be written as:

Thus, the new average is $\frac{1}{2}x_0 + 12$.

Explanation

Let:

• $P(h_1) =$ Probability of black in box 1
• $P(h_2) =$ Probability of black in box 2
• $P(p_1) =$ Probability of white in box 1
• $P(p_2) =$ Probability of white in box 2

Given the probability of getting black balls from both boxes is $\frac{3}{10}$. Since taking balls from box 1 and box 2 are independent events:

We need to find a pair of factors of $120$ where both are less than or equal to $20$ (maximum number of balls). The possible pair is $12$ and $10$ (or vice versa). Let $n(h_1) = 12$ and $n(h_2) = 10$.

Then the number of white balls is:

• Box 1: $20 - 12 = 8$ white balls
• Box 2: $20 - 10 = 10$ white balls

The probability that both balls are white is:

Given $\frac{m}{n} = \frac{1}{5}$. So $m = 1$ and $n = 5$. The value of $m + n = 1 + 5 = 6$.

Explanation

The order of members in a set does not matter, so we use the concept of combinations.

The total number of subsets of a set with $n$ members is $2^n$. Given $n = 10$, the total number of subsets is:

We are asked to find the number of subsets with more than $3$ members. This means we are looking for the sum of subsets with 4, 5, 6, ..., 10 members.

An easier way is to use the complement principle: Total subsets minus subsets with $0$, $1$, $2$, or $3$ members.

The combination formula is:

Let's calculate them one by one.

Subsets with 0 members

Subsets with 1 member

Subsets with 2 members

Subsets with 3 members

The number of subsets with $\leq 3$ members is:

Thus, the number of subsets with more than $3$ members is:

So, the number of subsets of $A$ that have more than $3$ members is $848$.

Explanation

Based on the definition of the operation $\odot$, we will calculate the value of $(5 \odot 2) \odot 3$ step by step. First, we calculate the part inside the parentheses, which is $5 \odot 2$ with $a = 5$ and $b = 2$.

Next, we operate the result with $3$. Now we calculate $24 \odot 3$ with $a = 24$ and $b = 3$.

Thus, the value of $(5 \odot 2) \odot 3$ is $208$.

Explanation

Let $A$ be Ali's current age and $T$ be Tika's current age. Given that the ratio of Ali's age to Tika's age three years ago was $4:5$.

Three years from now, the ratio of their ages will be $5:6$.

From equations $(1)$ and $(2)$, we perform elimination to find the value of $A$. Multiply equation $(1)$ by $5$ and equation $(2)$ by $4$.

Ali's current age is $27$ years. Next, we substitute $A = 27$ into equation $(1)$ to find $T$.

Thus, Tika's current age is $33$ years. The difference in their ages is:

Explanation

The equation of line $h$ is $y = mx + c$. Given $y = 3$ and $x = -1$, then:

Thus, the line equation can be written as $y = mx + (m + 3)$. Since line $h$ intersects the graph $y = -x^2$ at two distinct points, then:

For the line to intersect at two distinct points, the discriminant must satisfy $D > 0$.

The roots are $m = -2$ and $m = 6$. By testing intervals, we obtain the solution set:

Explanation

Given the inequality:

Rearrange it into the standard quadratic form by moving $-4$ to the left side:

For the quadratic function to be positive for every real $x$ (positive definite), its graph must lie entirely above the $x$-axis. The conditions are:

1. The coefficient of $x^2$ must be positive ($a > 0$).
2. The discriminant must be negative ($D < 0$).

Condition 1: $a > 0$

Condition 2: $D < 0$

Recall that $D = b^2 - 4ac$. With $a = 2m + 1$, $b = -4(m + 1)$, and $c = m + 4$:

Divide both sides by $4$ to simplify:

The roots are $m = 0$ and $m = \frac{1}{2}$. Since the inequality is "<", the solution lies between the roots:

Conclusion

We find the intersection of conditions $(1)$ and $(2)$:

• Condition $(1)$: $m > -\frac{1}{2}$
• Condition $(2)$: $0 < m < \frac{1}{2}$

The intersection of both regions is:

Explanation

Let:

• $J_A =$ Distance traveled by Ali until they meet
• $J_T =$ Distance traveled by Tika until they meet
• $v_A =$ Ali's speed $= 60$ km/h
• $v_T =$ Tika's speed $= 70$ km/h

Since Ali and Tika depart at the same time, the time they take to meet is also the same. Let $t =$ time.

Then:

Since the total distance is $221$ km, then:

So the time required until they meet is:

Thus, they will meet at:

Explanation

Let:

• $\bar{x}_K =$ Class average score
• $n_L =$ Number of male students
• $n_P =$ Number of female students
• $\bar{x}_L =$ Average score of male students
• $\bar{x}_P =$ Average score of female students

Given:

• $\bar{x}_K = 7.5$
• $\frac{n_L}{n_P} = \frac{3}{4} \implies n_L = 0.75 n_P$
• $\bar{x}_P = 1.75 + \bar{x}_L$

The class average score can be obtained as follows:

Substitute the known values:

Therefore, the average score of female students is:

Explanation

The fewer workers there are, the longer it takes to complete the work. Thus, this is an inverse proportion.

Let $x =$ days required.

So, the time required to complete the order if only $15$ people are working is $16$ days. Therefore, the additional time required is:

Explanation

The number of ways to choose red balls is:

The number of ways to choose white balls is:

The total number of possible ways to choose the balls is:

Let $A =$ The event of drawing $2$ red balls and $1$ white ball.

The probability of drawing $2$ red balls and $1$ white ball is:

Explanation

In addition or subtraction operations, we can use the associative property to simplify the calculation.

There are $\frac{2014}{2} = 1007$ groups, so:

Explanation

The given number sequence is:

Let's analyze the difference between consecutive terms:

Observing the pattern of differences ($2.5, \ 5, \ 10, \ 20, \ 40$), we can see that each difference is obtained by multiplying the previous difference by $2$ (a common ratio of $2$).

Therefore, the next difference after $40$ is:

Thus, the number that fills the blank is:

To verify this pattern, let's calculate the subsequent term and check if it matches the last known term ($318$). The next difference after $80$ would be $80 \times 2 = 160$.

The calculation matches the last term in the problem.

Therefore, the correct number to complete the pattern is $158$.

Explanation

The fractions can be rewritten in the form $1 + \frac{1}{x}$ as follows:

So, all numbers have the form $1 + \frac{1}{x}$. To find the largest value, we need to find the fraction where $\frac{1}{x}$ is largest. The value of $\frac{1}{x}$ increases as the denominator $x$ decreases.

From the five fractions above, the values of $x$ are $33, 60, 55, 124,$ and $25$ respectively. The smallest $x$ is $25$.

Therefore, the largest value is $\frac{26}{25}$.

Let's calculate its decimal value:

The decimal value is $1.040$.

Explanation

Since the water in the hemispherical container is transferred to the cylindrical container, the volume of water remains the same.

Since the radius of the sphere is the same as the radius of the cylinder, we can simplify the equation by dividing both sides by $\pi \cdot r^2$:

Given that the radius $r = 15 \text{ cm}$. We substitute this value into the equation:

Thus, the height of the water in the cylinder is $10 \text{ cm}$.

Explanation

Let:

• $n(S)$ = Total number of students in the class
• $n(P)$ = Number of students in scouts
• $n(F)$ = Number of students in futsal

Given:

The number of students who do not participate in either ($n(P \cup F)^c$) can be found using the universal set formula:

Substitute the known values:

Thus, the number of students who do not join scouts and futsal extracurriculars is $10$.