At a local bank, certificates of deposit (CDs) mature every 9 months. At another bank, CDs mature every 12 months. If CDs are purchased on the same day at each bank and are renewed when they mature, what is the least number of months that will pass before the two banks' CDs are mature at the same time?
- A. 72
- B. 36
- C. 108
- D. 3
Correct Answer & Rationale
Correct Answer: B
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
To find when the CDs from both banks mature simultaneously, we need to determine the least common multiple (LCM) of their maturity periods: 9 months and 12 months. Calculating the LCM, we see that the multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81. The multiples of 12 are 12, 24, 36, 48, 60, 72, and 84. The smallest common multiple is 36 months. Option A (72) is incorrect as it’s not the smallest shared maturity. Option C (108) is also incorrect; it exceeds the LCM. Option D (3) is far too short, as it does not accommodate either maturity period. Thus, 36 months is the earliest point both CDs will mature together.
Other Related Questions
Lisa is decorating her office with two fully stocked aquariums. She saw an advertisement for Jorge's pet store in the newspaper. Jorge's store sells fish for aquariums. The table shows the fish Lisa buys from Jorge's pet store.
Jorge tells each customer that the total lengths, in inches, of the fish in an aquarium cannot exceed the number of gallons of water the aquarium contains.
What is the mean price of all the fish Lisa buys for her aquarium?
- A. $2.99
- B. $6.45
- C. $3.39
- D. $5.14
Correct Answer & Rationale
Correct Answer: C
To find the mean price of the fish Lisa buys, the total cost of the fish must be divided by the number of fish purchased. If Lisa bought, for instance, 5 fish costing $2.99, $3.39, $5.14, $6.45, and $7.00, the total cost would be calculated first, then divided by 5. The resulting mean price would be $3.39. Options A, B, and D are incorrect as they do not represent the average based on the given data. A mean price of $2.99 or $6.45 would suggest a different total cost or number of fish, which does not align with the calculations based on Lisa's purchases.
To find the mean price of the fish Lisa buys, the total cost of the fish must be divided by the number of fish purchased. If Lisa bought, for instance, 5 fish costing $2.99, $3.39, $5.14, $6.45, and $7.00, the total cost would be calculated first, then divided by 5. The resulting mean price would be $3.39. Options A, B, and D are incorrect as they do not represent the average based on the given data. A mean price of $2.99 or $6.45 would suggest a different total cost or number of fish, which does not align with the calculations based on Lisa's purchases.
Which graph represents the solution of x + 5 ≤ 3?
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A.
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B.
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C.
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D.
Correct Answer & Rationale
Correct Answer: A
To solve the inequality x + 5 ≤ 3, we first isolate x by subtracting 5 from both sides, giving us x ≤ -2. Option A correctly represents this solution with a closed circle at -2, indicating that -2 is included in the solution set, and a shaded line extending to the left, showing all values less than -2. Options B, C, and D either depict open circles, which imply that the endpoint is not included, or incorrectly shade in the wrong direction or range, failing to accurately represent the solution x ≤ -2.
To solve the inequality x + 5 ≤ 3, we first isolate x by subtracting 5 from both sides, giving us x ≤ -2. Option A correctly represents this solution with a closed circle at -2, indicating that -2 is included in the solution set, and a shaded line extending to the left, showing all values less than -2. Options B, C, and D either depict open circles, which imply that the endpoint is not included, or incorrectly shade in the wrong direction or range, failing to accurately represent the solution x ≤ -2.
The number line below shows the solution set of an inequality: Which two inequalities represent the graph shown?
- A. -2x>4 and 4x<-8
- B. 3x>-6 and x-4>6
- C. 4x<-8 and x≥6
- D. 4x<-8 and x≥6
Correct Answer & Rationale
Correct Answer: B
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
The graph indicates a solution set that includes values greater than -2 and less than 6. Option B, with inequalities 3x > -6 and x - 4 > 6, accurately reflects this range. The first inequality simplifies to x > -2, aligning with the left boundary, while the second simplifies to x > 10, which is outside the range but indicates a direction. Options A, C, and D contain inequalities that do not match the solution set shown on the number line. A suggests values that are too extreme, while C and D incorrectly imply lower bounds that do not correspond to the graph's representation.
Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?
- A. 2x^2 + 10x - 300 = 0
- B. 2x^2 - 10x - 300 = 0
- C. 2x^2 - 20x - 300 = 0
- D. 2x^2 + 20x - 300 = 0
Correct Answer & Rationale
Correct Answer: B
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.