Intermediate Mathematics Flashcards

1. Two angles of a triangle each measure 70°. What is the measure of the third angle in degrees?

Answer: 40°

The fundamental property of any triangle is that the sum of its interior angles always equals 180 degrees. Given that two angles each measure 70°, their combined measure is 140°. To find the third angle, subtract this sum from 180° (180° - 140° = 40°), resulting in 40 degrees for the third angle.

2. If Jack needs 2 ½ pints of cream to make a dessert. How many pints will he need to make 3 desserts?

Answer: 7 ½

To find the total amount of cream needed, you multiply the amount required for one dessert by the number of desserts Jack wants to make. Jack needs 2 ½ pints (or 2.5 pints) for each dessert. Therefore, for 3 desserts, he will need 2.5 pints/dessert * 3 desserts = 7.5 pints, which is equivalent to 7 ½ pints.

3. A discount store takes 50% off of the retail price of a desk. For the store's holiday sale, it takes an additional 20% off of all furniture. The desk's retail price was $320. How much is the desk on sale for during the holiday sale?

Answer: $128

First, calculate the price after the initial 50% discount: 50% of $320 is $160, so the desk's price becomes $320 - $160 = $160. Next, apply the additional 20% holiday sale discount to this new price. 20% of $160 is $32. Subtracting this second discount gives the final sale price: $160 - $32 = $128.

4. Which vacation destination is most common for the students?

Answer: Beach

To determine the most common vacation destination, one would examine the provided graph or chart and identify the category with the highest frequency or percentage. Based on the correct answer, the "Beach" category must have represented the largest portion of students' vacation choices, indicating it was the most popular option.

5. If 500 students attend Washington Middle School, how many are going to the mountains for vacation?

Answer: 60

To calculate the number of students going to the mountains, you first need to identify the percentage of students choosing mountains from the accompanying chart or graph. If, for example, the chart indicates 12% of students go to the mountains, you would then calculate 12% of the total 500 students. This calculation (0.12 * 500) yields 60 students.

6. If a ¼ of a teaspoon is 1 ml, then how many milliliters are in 6 teaspoons?

Answer: 24 ml

First, determine how many milliliters are in one full teaspoon. Since ¼ of a teaspoon equals 1 ml, a full teaspoon is 4 times that amount (1 ml / (1/4) = 4 ml). Then, to find the total milliliters in 6 teaspoons, multiply 6 by 4 ml, which gives a total of 24 ml.

7. Which of the following is the correct graph for x≥3 or x≤ -2?

Answer: Line D

The inequality "x ≥ 3" is represented by a closed circle at 3 with an arrow extending to the right, indicating all numbers greater than or equal to 3. The inequality "x ≤ -2" is represented by a closed circle at -2 with an arrow extending to the left, indicating all numbers less than or equal to -2. Since the condition is "or," the correct graph (Line D) displays both of these separate rays on the number line.

8. A scale on a map states that every ¼ of an inch represents 20 miles. If two cities are 3 ½ inches apart, how many miles are actually between the two cities?

Answer: 280 miles

First, convert the given distance of 3 ½ inches to a decimal, which is 3.5 inches. Since every ¼ inch represents 20 miles, determine how many ¼-inch segments are in 3.5 inches by dividing 3.5 by 0.25, which equals 14. Finally, multiply this number by the mileage represented by each segment: 14 * 20 miles = 280 miles.

9. Michelle wants to expand her flowerbed by increasing the length and width each by 2 ft. What will the new area of the flowerbed be, if L and W represent the original dimensions of the flowerbed's length and width?

Answer: (L+2) (W+2)

The original dimensions of the flowerbed are represented by L for length and W for width. When Michelle increases the length by 2 ft, the new length becomes (L+2). Similarly, increasing the width by 2 ft results in a new width of (W+2). The area of a rectangle is calculated by multiplying its length and width, so the new area of the expanded flowerbed will be (L+2) multiplied by (W+2).

10. Melinda's lights went out. She has 3 pairs of red socks in her drawer, 2 pairs of black socks, and 5 pairs of white socks. What is the minimum number of pairs she must remove from the drawer to ensure that she has a pair of each color?

Answer: 9

This is a classic "worst-case scenario" problem. To guarantee she has at least one pair of each color, Melinda must first draw all the pairs of the two most numerous colors. She has 5 white and 3 red pairs, so she could draw all 5 white pairs and all 3 red pairs (5 + 3 = 8 pairs) without getting a black pair. The very next pair she draws (the 9th pair) is guaranteed to be black, ensuring she has one of each color.

11. Which of the following fractions are correctly placed from the least in value to the greatest in value?

Answer: 1/4, 17/25, 11/16, 3/4

To correctly order the fractions from least to greatest, convert each fraction into its decimal equivalent. 1/4 is 0.25, 17/25 is 0.68, 11/16 is 0.6875, and 3/4 is 0.75. Arranging these decimals in ascending order gives 0.25, 0.68, 0.6875, and 0.75, which corresponds to the original fractions 1/4, 17/25, 11/16, and 3/4 respectively.

12. What is the mathematical average of the number of days in a typical year, the number of days in a week, and the number of hours in a day?

Answer: 132

To find the mathematical average, first identify the three numbers: a typical year has 365 days, a week has 7 days, and a day has 24 hours. Next, sum these three values: 365 + 7 + 24 = 396. Finally, divide the sum by the count of numbers (which is 3) to get the average: 396 / 3 = 132.

13. 1.75 x 105=

Answer: 175,000

Scientific notation 1.75 x 105 indicates that the decimal point in 1.75 should be moved 5 places to the right. Starting with 1.75, moving the decimal five places to the right results in 175,000. Each move to the right effectively multiplies the number by 10.

14. The electric company charges 3 cents per kilowatt-hour. George used 2800 kilowatt-hours in April, 3200 kilowatt-hours in May, and 3600 kilowatt-hours in June. What was his average cost of electricity for the 3 months?

Answer: $96

First, calculate the total kilowatt-hours used over the three months: 2800 + 3200 + 3600 = 9600 kWh. Next, determine the total cost by multiplying the total kilowatt-hours by the rate of 3 cents ($0.03) per kilowatt-hour: 9600 * $0.03 = $288. Finally, divide the total cost by the number of months (3) to find the average cost per month: $288 / 3 = $96.

15. On a map, 1/3 inch equals 15 miles. The distance between two towns on a map is 3 2/3 inches. How many miles are actually between the two towns?

Answer: 165

First, determine the mileage represented by one full inch. Since 1/3 inch equals 15 miles, one inch would be 3 times that distance, or 45 miles (15 * 3). Next, convert the distance between the towns, 3 2/3 inches, into an improper fraction, which is 11/3 inches. Finally, multiply the mileage per inch by this fraction: (11/3) * 45 miles = 11 * 15 miles = 165 miles.

16. James invested $4,000 at 5% interest per year; how long will it take him to earn $200 in simple interest?

Answer: 1 year

To find the time it takes to earn $200 in simple interest, use the formula I = PRT (Interest = Principal × Rate × Time). We have I = $200, P = $4000, and R = 5% or 0.05. Plugging these values in gives $200 = $4000 * 0.05 * T. This simplifies to $200 = $200 * T, so dividing both sides by $200 reveals that T = 1 year.

17. John pays $650 in property tax. What is the assessed value of his property if property taxes are 1.2% of assessed value?

Answer: $54,166.67

To find the assessed value, you need to set up an equation where the property tax is 1.2% of the assessed value. Let 'A' represent the assessed value. So, $650 = 0.012 * A. To solve for A, divide the property tax by the tax rate: A = $650 / 0.012. This calculation yields an assessed value of approximately $54,166.67.

18. A lamp is marked with a sale price of $23.80, which is 15% off of the regular price. What is the regular price?

Answer: $28

If the lamp is on sale for 15% off, the sale price of $23.80 represents 85% (100% - 15%) of the original regular price. To find the regular price, divide the sale price by the percentage it represents in decimal form. So, $23.80 / 0.85 equals $28, which is the lamp's original regular price.

19. A mattress store sells their stock for 15% off of retail. If someone pays cash, they take an additional 10% off of the discounted price. If a mattress's retail price is $750, what is the price after the store discount and the cash discount?

Answer: $573.75

First, calculate the price after the initial 15% store discount. 15% of $750 is $112.50, making the discounted price $750 - $112.50 = $637.50. Next, apply the additional 10% cash discount to this new discounted price. 10% of $637.50 is $63.75. Subtracting this second discount gives the final price: $637.50 - $63.75 = $573.75.

20. 85% of what number is 136?

Answer: 160

To find the original number, convert the percentage to a decimal by dividing it by 100 (85% = 0.85). Then, set up the equation: 0.85 multiplied by the unknown number (x) equals 136. Dividing 136 by 0.85 gives the result of 160.