Math Flashcards

Frank measured the amount of liquid that evaporated over a 12-day period from his container with l equaling ounces of liquid. At the end of the day too, it had lost 3 ounces. At the end of day seven, an additional 6 ounces. By the end of day 12, the container had lost one-third of what remained at the end of day seven. Which of the following represents the remaining amount of liquid in ounces in Frank's container at the end of day 12?

Answer: l-9/3

First, calculate the total amount of liquid lost by the end of day seven: 3 ounces on day 2 + 6 additional ounces by day 7 = 9 ounces lost. This leaves l - 9 ounces remaining at the end of day seven. The problem states that by the end of day 12, the container had lost one-third of what remained at the end of day seven. This implies that one-third of the amount (l-9) *is* the amount remaining at day 12. Therefore, the remaining amount is (l - 9) / 3.

An entertainment superstore is running a special sale on DVDs and Blu-ray. DVDs (d) are priced at $8 during the sale while Blu-ray (b) are priced at $15. Each price point is a 20% reduction off the regular price. The sale pulls in $1,250 with 60 total units sold. During a regular day, DVDs and Blu-ray combine for around $800 in total sales while moving two-thirds the number of units on the special sale day. Solving which of the following systems of equations will reveal d and b sold during a regular day?

Answer: d+b=4010d+18.75b=800

The question asks for the system of equations representing a *regular day*. First, calculate the number of units sold on a regular day: (2/3) * 60 sale units = 40 units, so d + b = 40. Next, determine the regular prices: since sale prices are 20% off, they are 80% of the regular price. Regular DVD price = $8 / 0.80 = $10. Regular Blu-ray price = $15 / 0.80 = $18.75. Finally, the total sales on a regular day are $800, so 10d + 18.75b = 800. These two equations form the correct system.

Blake earns base pay of $10 per hour at a clothing boutique. He also earns a certain percentage commission on sales. Blake worked an average of 37 hours per week over a two-week period. If he also helped broker $7,500 in sales and his total pay is $1,000 for the two-week period, approximately what percentage commission does he earn?

Answer: 3.5%

First, calculate Blake's base pay for the two-week period: 37 hours/week * 2 weeks * $10/hour = $740. His total pay was $1,000, so the amount earned from commission is $1,000 - $740 = $260. To find the commission percentage, divide the commission earned by the total sales: $260 / $7,500 = 0.03466... Multiplying by 100 to convert to a percentage gives approximately 3.47%, which rounds to 3.5%.

If (C+x)x-3=x+83, which of the following could be an expression of C in terms of x?

Answer: C=13(x+6)(x-4)

To solve for C, first cross-multiply the given equation: 3(C+x) = (x-3)(x+8). Expand the right side: 3C + 3x = x^2 + 8x - 3x - 24, which simplifies to 3C + 3x = x^2 + 5x - 24. Next, subtract 3x from both sides: 3C = x^2 + 2x - 24. Finally, divide by 3: C = (x^2 + 2x - 24) / 3. Factoring the quadratic in the numerator (finding two numbers that multiply to -24 and add to 2, which are 6 and -4) gives C = (x+6)(x-4)/3, or C = 1/3(x+6)(x-4).

Larry wants to complete a marathon in under 6 hours, but he knows he can't run the 26.2 miles the whole way. He'll need to run (r) and walk (w). If he is able to walk at four miles per hour and run at six miles per hour, which system of inequalities shows his total running and walking time?

Answer: 4w+6r>26.2w+r6

The problem requires two inequalities. The first concerns the total time: Larry wants to finish "under 6 hours," so his total walking time (w) plus running time (r) must be less than 6 hours (w + r 26.2 is the closest representation, implying he must cover the distance successfully.

In a crossbow manufacturing facility, a quality control expert tests a randomly selected group of 1000 crossbows. If the expert finds 17 of the randomly selected crossbows are defective, which of the following inferences would be most supported?

Answer: 98.3% of the crossbows are not defective.

In the randomly selected group of 1000 crossbows, 17 were found to be defective. To find the number of non-defective crossbows, subtract the defective ones from the total: 1000 - 17 = 983. To express this as a percentage, divide the number of non-defective crossbows by the total sample size and multiply by 100: (983 / 1000) * 100% = 98.3%. This calculation directly supports the inference that 98.3% of the crossbows are not defective.

If -196-136, what is the greatest possible integer value of 16z-28?

Answer: 12

To find the greatest possible integer value of 16z-28, we can manipulate the given inequality directly. Multiply all parts of the inequality -19/6 < -4z+7 < -13/6 by -4. Remember to reverse the inequality signs when multiplying by a negative number, resulting in -4(-13/6) < -4(-4z+7) < -4(-19/6). This simplifies to 52/6 < 16z-28 < 76/6. Converting these fractions to decimals, we get approximately 8.67 < 16z-28 < 12.67. The greatest integer value that satisfies this range is 12.

Which of the following ordered pairs (j,k) is the solution to the system of equations below? -12-4j=6k4+74k=-87j

Answer: (0,-2)

To determine the solution to a system of equations, an ordered pair (j,k) must satisfy both equations simultaneously. We can test the given options by substituting the values of j and k into each equation. For the ordered pair (0,-2), substituting j=0 and k=-2 into the first equation, -12-4j=6k, yields -12-4(0)=6(-2), which simplifies to -12=-12, confirming it satisfies the first equation. However, substituting these values into the second equation, 4+(7/4)k=(-8/7)j, yields 4+(7/4)(-2)=(-8/7)(0), simplifying to 4-7/2=0, or 1/2=0, which is false. Therefore, while (0,-2) satisfies the first equation, it does not satisfy the second, meaning it is not the solution to the system as written.

In the expression 76123y-5y=1, what is the value of y?

Answer: -67

The given equation is a complex fraction equal to 1, which means the numerator must be equal to the denominator. So, we set 7/6 equal to (12/3y) - (5/y). First, simplify the term 12/3y to 4/y. Then, combine the terms on the right side of the equation: 4/y - 5/y = -1/y. This simplifies the equation to 7/6 = -1/y. Finally, cross-multiply to solve for y, yielding 7y = -6, which means y = -6/7.