Tüm alıştırma soruları

An artisanal coffee roaster creates a custom blend using Arabica, Robusta, and Liberica beans in the initial ratio of $5 : 3 : 2$ by weight, respectively. To adjust the flavor profile, the roaster adds $60$ kilograms of Arabica beans and $20$ kilograms of Robusta beans to the blend, while leaving the amount of Liberica beans unchanged. If the new ratio of Arabica to Robusta to Liberica beans is $4 : 2 : 1$, respectively, what was the initial total weight, in kilograms, of the blend?

A municipal water treatment facility utilizes three primary purification units—Unit P, Unit Q, and Unit R—to process large water batches. Operating alone at its constant rate, Unit P can process a full batch in $12$ hours, Unit Q can process a full batch in $18$ hours, and Unit R can process a full batch in $36$ hours. All three units begin processing a batch together. After $3$ hours, Unit P stops operating due to scheduled maintenance. Unit Q and Unit R continue operating together for an additional $2$ hours, after which Unit Q is shut down. Unit R then completes the remainder of the batch working alone. How many total hours does it take from start to finish to process the complete batch?

If $m$ and $n$ are non-zero real numbers such that $m$ and $n$ are the roots of the quadratic equation $x^2 + mx + n = 0$, what is the value of $m - n$?

A clinical study evaluated $150$ patient records for the presence of three specific health biomarkers: Biomarker X, Biomarker Y, and Biomarker Z. The study revealed that $70$ patients had Biomarker X, $60$ patients had Biomarker Y, and $50$ patients had Biomarker Z. Exactly $35$ patients had exactly two of these biomarkers, and exactly $10$ patients had all three biomarkers. How many of the patients evaluated had none of the three biomarkers?

If $|2x - 5| = 9$, what is the sum of all possible values of $x$?

A manufacturing plant has a fixed daily setup cost of $\$1,500$. For a specific product, the production cost for each of the first $50$ units is $\$60$ per unit, while each additional unit produced beyond the first $50$ costs $\$30$ per unit. If the average total cost per unit produced on a given day was $\$45$, how many total units were produced on that day?

Four distinct books—two mathematics books and two history books—are to be arranged in a single line on a shelf. If the two mathematics books must stand next to each other, in how many different linear arrangements can the four books be placed?

In an effort to reduce operating expenses, a major maritime freight carrier replaced its entire fleet of traditional steel shipping containers with newly engineered carbon-fiber containers that are 25 percent lighter. The carrier anticipated a significant drop in fuel consumption because vessel displacement was reduced while payload volume per voyage remained identical. However, during the year following the complete container replacement, the average quantity of fuel consumed per ton-mile of freight transported across the carrier's ocean routes increased by nearly 6 percent. Which of the following, if true, most helps to resolve the apparent paradox described above?

A sequence of positive integers $a_n$ is defined by $a_n = 2^n + 5^n$ for all integers $n \ge 1$. What is the remainder when $a_{100}$ is divided by $7$?

A set $S$ consists of consecutive integers. The arithmetic mean of all the positive integers in set $S$ is $18.5$, and the arithmetic mean of all the negative integers in set $S$ is $-12$. How many integers are in set $S$?

An arithmetic sequence $a_1, a_2, a_3, \dots$ with a positive common difference $d$ and a geometric sequence $b_1, b_2, b_3, \dots$ with a positive common ratio $r$ both have the same positive first term ($a_1 = b_1 > 0$). If the 3rd term of the arithmetic sequence equals the 3rd term of the geometric sequence ($a_3 = b_3$), and the 7th term of the arithmetic sequence equals the 5th term of the geometric sequence ($a_7 = b_5$), what is the value of $r$?

A manufacturing plant uses three assembly lines, $L_1$, $L_2$, and $L_3$, to produce three custom components, $X$, $Y$, and $Z$.

- Producing one unit of component $X$ requires 2 hours on $L_1$, 1 hour on $L_2$, and 3 hours on $L_3$.
- Producing one unit of component $Y$ requires 3 hours on $L_1$, 4 hours on $L_2$, and 2 hours on $L_3$.
- Producing one unit of component $Z$ requires 1 hour on $L_1$, 2 hours on $L_2$, and 4 hours on $L_3$.

During a given week, assembly lines $L_1$, $L_2$, and $L_3$ were operated for a total of 140 hours, 165 hours, and 235 hours, respectively, with zero idle time. Assuming full capacity utilization, how many units of component $Z$ were produced during that week?

A tech firm observed that employees who voluntarily chose to work remotely reported higher job satisfaction than those who worked in the office. Based on this observation, management concluded that transitioning all employees to remote work will increase overall employee job satisfaction across the company. Which of the following, if true, most seriously weakens management's conclusion?

Set $S$ consists of $n$ consecutive even integers. The arithmetic mean of all the integers in set $S$ is $45$. If the difference between the largest integer and the smallest integer in set $S$ is $34$, what is the value of the largest integer in set $S$?

Four numerical quantities $K$, $L$, $M$, and $N$ are defined based on arithmetic and geometric sequences as follows. Arrange these four quantities in ascending order (from smallest to largest value):

- **Quantity $K$**: The sum of the first 5 terms of an arithmetic sequence with first term $a_1 = 2$ and common difference $d = 3$.
- **Quantity $L$**: The 4th term of a geometric sequence with first term $b_1 = 3$ and common ratio $r = 2$.
- **Quantity $M$**: The sum of an infinite geometric series with first term $c_1 = 18$ and common ratio $r = \frac{1}{2}$.
- **Quantity $N$**: The 7th term of an arithmetic sequence with first term $d_1 = 50$ and common difference $d = -4$.