MATH Flashcards

The result of subtracting a number x from 36 and dividing the difference by x is 2. What does x represent?

Answer: 12

To solve this, we translate the problem into an algebraic equation. 'Subtracting a number x from 36' is (36 - x), and 'dividing the difference by x' is (36 - x) / x. Setting this equal to 2 gives the equation (36 - x) / x = 2. Multiplying both sides by x yields 36 - x = 2x. Adding x to both sides results in 36 = 3x, and finally, dividing by 3 gives x = 12.

Angela has a 6-mile-per-hour running speed. How many miles can she run in 90 minutes if she keeps the same pace?

Answer: 9

To find the distance Angela can run, first convert the time from minutes to hours: 90 minutes is equal to 1.5 hours. Then, use the formula Distance = Speed × Time. With a speed of 6 miles per hour and a time of 1.5 hours, the distance covered is 6 × 1.5 = 9 miles.

How many positive integers under 100 are divisible by three, five, and seven?

Answer: None of the above

For a number to be divisible by three, five, and seven, it must be a multiple of their least common multiple (LCM). Since 3, 5, and 7 are all prime numbers, their LCM is simply their product, which is 3 × 5 × 7 = 105. As we are looking for positive integers under 100, and the smallest such multiple is 105, there are no integers that satisfy the condition.

Which of the following statements regarding 0.6666 is correct?

Answer: It is a rational number.

A rational number is any number that can be expressed as a simple fraction, p/q, where p and q are integers and q is not zero. The decimal 0.6666... is a repeating decimal, which can be precisely represented as the fraction 2/3. Since it can be written as a ratio of two integers, it is by definition a rational number.

What number is eight times the largest negative integer?

Answer: 6

The largest negative integer is -1. Multiplying this by eight yields 8 * (-1) = -8. However, the provided correct answer is 6, which does not align with the direct calculation from the question's statement. This indicates a discrepancy between the question's phrasing and the designated correct option, as -8 is not among the choices and 6 cannot be derived from the given information.

What is the product of the four largest negative integers and the smallest positive integer?

Answer: m + n + 1

The four largest negative integers are -1, -2, -3, and -4. The smallest positive integer is 1. The product of these numbers is (-1) × (-2) × (-3) × (-4) × 1 = 24. The provided options are algebraic expressions (m+n+2, mn, m+n, mn+1, m+n+1) and do not represent a numerical value, indicating a significant mismatch between the question and the given answer choices.

Jack purchased purple tomatoes for $8 apiece from a premium vegetable stall, and Carol purchased organic huge cabbages for $12 each. They spent a total of $104 on ten vegetables. Carol bought how many cabbages?

Answer: 6

Let 't' be the number of tomatoes and 'c' be the number of cabbages. We can set up a system of two linear equations: 8t + 12c = 104 (total cost) and t + c = 10 (total number of vegetables). From the second equation, t = 10 - c. Substituting this into the first equation gives 8(10 - c) + 12c = 104. Simplifying, we get 80 - 8c + 12c = 104, which leads to 80 + 4c = 104. Solving for c, we find 4c = 24, so Carol bought 6 cabbages.

Between the numbers -10 and 10, how many even integers are there?

Answer: 9

The phrase 'between the numbers -10 and 10' means we include integers greater than -10 and less than 10. The even integers in this range are -8, -6, -4, -2, 0, 2, 4, 6, and 8. Counting these values, there are a total of nine even integers.

For a one-time price of $50 + $8 per sandwich, a caterer will prepare cabbage sandwiches for special events. In dollars, how much does an order of c cabbage sandwiches cost?

Answer: 50+8c

The total cost for the catering service consists of a fixed one-time price and a variable cost per sandwich. The one-time price is $50. For 'c' cabbage sandwiches, the variable cost is $8 multiplied by 'c', or 8c. Therefore, the total cost in dollars for an order of 'c' cabbage sandwiches is represented by the expression 50 + 8c.

A total of x dollars will be spent on 30 identical erasers. What is the total cost of 70 erasers in terms of x at this rate?

Answer: 7y/3

To find the cost of 70 erasers, first determine the unit cost: if 30 erasers cost 'y' dollars (assuming 'x' in the question is a typo and should be 'y' to match the answer options), then each eraser costs y/30 dollars. Multiplying this unit cost by 70 gives the total cost for 70 erasers: (y/30) * 70 = 70y/30, which simplifies to 7y/3. This proportional reasoning allows us to calculate the cost of a different quantity of erasers at the same rate.

Six purple marshmallows are included in a carton of only purple and green marshmallows. How many green marshmallows are in the box if the likelihood of selecting a purple marshmallow from the box is 1/3?

Answer: 12

The probability of selecting a purple marshmallow is given as 1/3, and there are 6 purple marshmallows. We can set up the equation: (Number of Purple) / (Total Number) = 1/3. Substituting the known values, 6 / Total Number = 1/3. Solving for the total number of marshmallows, we get Total Number = 6 * 3 = 18. Since there are 18 marshmallows in total and 6 are purple, the number of green marshmallows is 18 - 6 = 12.

When a number is divided by four. The result is divided by three, resulting in a final score of two. What was the original number?

Answer: 24

To find the original number, we need to reverse the operations in the opposite order. The final result is 2. Since this result was obtained by dividing by three, the number before that division must have been 2 multiplied by 3, which is 6. This number (6) was obtained by dividing the original number by four, so the original number must have been 6 multiplied by 4, which equals 24.

What is the sum of the smallest and greatest primes that is less than ten?

Answer: 9

To find the sum, first identify the prime numbers less than ten. These are 2, 3, 5, and 7. The smallest prime number in this set is 2, and the greatest is 7. Adding these two values together, 2 + 7, gives a sum of 9.

Which of the following might be a prime number if p is a prime number?

Answer: p + 2

A prime number is a whole number greater than 1 with no divisors other than 1 and itself. If 'p' is a prime number, options like 2p (always even and greater than 2 for p>1) or p squared (always has factors 1, p, p^2) cannot be prime. The expression p+2, however, can result in a prime number for certain prime values of p (e.g., if p=3, p+2=5, which is prime), making it the only option that *might* be a prime number.

What is the sum of the lowest prime and the largest negative even integer?

Answer: 0

The lowest prime number is 2, as 1 is not considered prime and 2 is the smallest integer divisible only by 1 and itself. The largest negative even integer is -2, as even integers are multiples of 2 and -2 is the largest among the negative ones. Summing these two values, 2 + (-2), results in 0.

The remainder of dividing the positive integer m by 7 is 4. What is the remainder after dividing m + 26 by 7?

Answer: 2

If a positive integer 'm' has a remainder of 4 when divided by 7, we can write m as 7k + 4 for some integer k. We need to find the remainder of (m + 26) when divided by 7. Substituting m, we get (7k + 4 + 26), which simplifies to 7k + 30. Since 7k is perfectly divisible by 7, we only need to find the remainder of 30 divided by 7. Dividing 30 by 7 gives 4 with a remainder of 2 (30 = 4 * 7 + 2).

In one hour, a farmer can collect 12 cabbages. How many hours would it take two farmers to gather 48 cabbages if they worked at the same rate?

Answer: 2

A single farmer collects 12 cabbages in one hour. If two farmers work at the same rate, their combined collection rate is 12 cabbages/hour * 2 farmers = 24 cabbages/hour. To gather 48 cabbages, the two farmers would take 48 cabbages / 24 cabbages/hour = 2 hours.

A specific rectangle has a surface area of 72. What is the width of the rectangle if the length of the rectangle is twice the width?

Answer: 6

The area of a rectangle is calculated by multiplying its length by its width (Area = L * W). Given the area is 72 and the length (L) is twice the width (W), we can write L = 2W. Substituting this into the area formula gives (2W) * W = 72, which simplifies to 2W^2 = 72. Dividing by 2, we get W^2 = 36, so the width W is the square root of 36, which is 6.

In the following equation, what is the value of x? 10+4(𝑥+5−5𝑥) = 30

Answer: 12

To solve the equation 10 + 4(x + 5 - 5x) = 30, first simplify the expression inside the parentheses: x + 5 - 5x becomes 5 - 4x. The equation then becomes 10 + 4(5 - 4x) = 30. Distribute the 4: 10 + 20 - 16x = 30, which simplifies to 30 - 16x = 30. Subtracting 30 from both sides gives -16x = 0, so x = 0. (Note: There appears to be a discrepancy between the provided question and the correct answer. If the question intended a different value on the right side, such as -162, then x=12 would be the solution.)

What is the perimeter of a square with a 64-square-inch surface area?

Answer: 32

The surface area of a square is found by squaring the length of one of its sides (Area = side^2). Given the area is 64 square inches, the side length is the square root of 64, which is 8 inches. The perimeter of a square is found by multiplying the side length by 4 (Perimeter = 4 * side). Therefore, the perimeter is 4 * 8 inches = 32 inches.