A pound of tea makes 210 cups of tea, and a pound of coffee makes 40 cups of coffee. If, on the average, a restaurant serves 12 times as many cups of coffee as tea, what is the ratio of pounds of tea to pounds of coffee used at the restaurant?
Answer(s): A
Let T be the number of cups of tea served.Then the number of cups of coffee served is 12T (since coffee is served 12 times as much as tea).Since 1 pound of tea makes 210 cups, the pounds of tea used is:Since 1 pound of coffee makes 40 cups, the pounds of coffee used is:Find the Ratio of Pounds of Tea to Coffee:
Pat has five matched pairs of socks and no two of the pairs are the same color. If Pat selects two socks simultaneously and at random, what is the probability that the two socks selected will be a matched pair?
Answer(s): D
Pat has 10 socks in total (5 pairs, each with 2 socks). The total number of ways to select 2 socks from 10 is given by the combination formula:Since there are 5 matched pairs of socks, there are 5 ways to select one matched pair (one for each of the 5 pairs).The probability of selecting a matched pair is the ratio of the number of ways to select a matched pair to the total number of ways to select 2 socks:
Two boats depart from the same point at the same time. Boat X travels north at a constant speed of 4 miles per hour while boat Y travels east at a constant speed of 3 miles per hour. In how many hours will the two boats be 10 miles apart?
Answer(s): C
To solve this, we can use the Pythagorean theorem, since the boats are traveling at right angles to each other (north and east).Let t be the time in hours. At time t, the distance traveled by Boat X (traveling north) is:Distance by Boat X = 4tmiles.Similarly, the distance traveled by Boat Y (traveling east) is:Distance by Boat Y = 3tmilesSince the two boats are traveling at right angles to each other, we can apply the Pythagorean theorem to find the distance between them at time t:We are given that the distance between the boats is 10 miles, so:
A volume of 30 cubic feet is equal to approximately how many gallons? (1 foot = 12 inches, and 1 gallon = 231 cubic inches.)
Answer(s): B
First, convert the volume from cubic feet to cubic inches. Since 1foot = 12inches, we have:1cubic foot = (12inches)3 = 12 × 12 × 12 = 1,728cubic inchesThus, 30 cubic feet is:30cubic feet = 30 × 1,728 = 51,840cubic inchesSince 1 gallon = 231 cubic inches, we can convert the volume in cubic inches to gallons:Thus, 30 cubic feet is approximately 225 gallons.
Total revenue from the sale of 5,100 concert tickets was $205,000. If each ticket was sold for either $30 or $50, what was the revenue from the sale of the $50 tickets?
Answer(s): E
Let the number of $30 tickets sold be denoted as x and the number of $50 tickets sold be denoted as y.1. The total number of tickets sold is 5,100:x + y = 5,100The total revenue from the sale of the tickets is $205,000. The revenue from selling x tickets at $30 each and y tickets at $50 each is:30x + 50y = 205,000x = 5,100 - y30(5,100 - y) + 50y = 205,00030 × 5,100 - 30y + 50y = 205,000153,000 - 30y + 50y = 205,000153,000 + 20y = 205,00020y = 52,000The number of $50 tickets sold is y = 2,600, and the revenue from these tickets is:50 × 2,600 = 130,000
Irene had $20,000 in savings. She used some of her savings to purchase a computer. Of the amount remaining, she invested 40 percent at 6 percent simple annual interest, and the rest at 10 percent simple annual interest. If Irene's income from the interest totaled $1,260 for the first year, how much of her savings did she use to purchase the computer?
Let x be the amount of Irene's savings that she used to purchase the computer. This means that the remaining savings after purchasing the computer is 20,000 - x.Irene invested 40 percent of the remaining savings at 6 percent interest. The amount invested at 6 percent is 0.4(20,000 - x).The remaining 60 percent of the remaining savings was invested at 10 percent interest. The amount invested at 10 percent is 0.6(20,000 - x).The interest earned from each investment can be calculated using the formula for simple interest:Interest = Principal × Rate × TimeIn this case, the time is 1 year.The interest from the 6 percent investment is:0.4(20,000 - x) × 0.06The interest from the 10 percent investment is:0.6(20,000 - x) × 0.10The total interest Irene earned is $1,260, so we can set up the following equation:0.4(20,000 - x) × 0.06 + 0.6(20,000 - x) × 0.10 = 1,2600.4(20,000 - x) × 0.06 = 0.024(20,000 - x)0.6(20,000 - x) × 0.10 = 0.06(20,000 - x)0.024(20,000 - x) + 0.06(20,000 - x) = 1,260(0.024 + 0.06)(20,000 - x) = 1,2600.084(20,000 - x) = 1,260x = 20,000 - 15,000 = 5,000
The cost of producing 1 unit of a product at a certain manufacturing company is $0.75. According to a sales model, if each unit has a selling price of $1.00, the company expects to sell 1,000 units each month. For each $0.05 increase in the unit selling price, the company expects to see a decrease of 100 units sold each month. According to the model, which of the following could be a unit selling price at which the company would expect a profit of at least $250.00 each month? (Profit equals revenue minus cost.) (Choose all that apply.)
Answer(s): A,B,C
Revenue is given by the selling price per unit multiplied by the number of units sold.Cost is given by the cost per unit multiplied by the number of units produced.Let the unit selling price be represented by p. If the price increases by $0.05, the number of units sold decreases by 100. Thus, the number of units sold can be written as:Profit = Revenue - CostWe want the profit to be at least $250. Let's plug in the equations for revenue and cost and solve for p:For p = 1.05:For p = 1.10:For p = 1.20:For p = 1.30:For p = 1.35:The valid unit selling prices at which the company would expect a profit of at least $250.00 are $1.05, $1.10, and $1.20.
A company has assets worth $150,000 and liabilities worth $70,000, giving it an asset-to-liability ratio of approximately 2.1. The company will borrow x dollars, and the amount borrowed will be added to both the assets and the liabilities. If the asset-to-liability ratio is to be greater than 1.2 after the money is borrowed, which of the following could be the value of x? (Choose all that apply.)
Answer(s): A,B
The initial asset-to-liability ratio is given as approximately 2.1, with assets of $150,000 and liabilities of $70,000.The formula for the asset-to-liability ratio is:After the company borrows x dollars, both the assets and liabilities increase by x. So:New assets = 150,000 + xNew liabilities = 70,000 + xWe are told that the asset-to-liability ratio must be greater than 1.2:To solve for x, we first multiply both sides of the inequality by 70,000 + x to eliminate the denominator (assuming 70,000 + x > 0):150,000 + x > 1.2(70,000 + x)150,000 + x > 84,000 + 1.2x150,000 - 84,000 > 1.2x - x66,000 > 0.2xWe have determined that x must be less than 330,000 to satisfy the condition that the asset-to-liability ratio is greater than 1.2. Now, we check the options:300,000: This value is less than 330,000, so it is a valid solution.320,000: This value is less than 330,000, so it is a valid solution.340,000: This value is greater than 330,000, so it is not a valid solution.360,000: This value is greater than 330,000, so it is not a valid solution.
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