Susan spent one-third of her money on books and half of the remaining money on clothing. She then spent three-fourths of what she had left on food. She had $5 left over. How much money did she start with?
Answer(s): A
Start with the money she had left and work backwards. If she had $5 left over, and had just spent three-fourths of her money on food, then $5 must be one-fourth of her money. Before buying food she must have had 5 × 4 = $20. She then spent half of her money on clothes; therefore, $20 was half of her money, giving her $40 at this point. She then spent one-third of her money on books and had $40 left over. If $40 represents two-thirds of her money, then $60 must be the amount she began with.
A truck travels 20 miles due north, 30 miles due east, and then 20 miles due north. How many miles is the truck from the starting point?
Answer(s): D
Draw a diagram to show the path of the truck.The distance between the starting point and the final destination is a diagonal line. This line is the hypotenuse of a right triangle that has one leg of 40 and the other measuring 30. Use the Pythagorean theorem: a2 + b2 = c2. Recall, however, that this is a multiple of the most common Pythagorean triple (3, 4, 5) — namely, 30, 40, 50. The distance is 50 miles.
A rectangular swimming pool is 20 feet by 28 feet. A deck that has uniform width surrounds the pool. The total area of the pool and deck is 884 square feet. What is the width of the deck?
Answer(s): C
Since we are trying to find the width of the deck, let x = the width of the deck. Therefore, x + x + 20 or 2x + 20 is the width of the entire figure. In the same way, x + x + 28 or 2x + 28 is the length of the entire figure.The area of a rectangle is length × width, so use A = l × w.Substitute into the equation: 884 = (2x + 20)(2x + 28)Multiply using FOIL: 884 = 4x2 + 56x + 40x + 560Combine like terms: 884 = 4x2 + 96x + 560Subtract 884 from both sides: 884 – 884 = 4x2 + 96x + 560 – 8840 = 4x2 + 96x – 324Divide each term by 4: 0 = x2 + 24x – 81Factor the trinomial: 0 = (x + 27)(x – 3)Set each factor equal to zero and solve: x + 27 = 0 or x – 3 = 0x = –27 x = 3Since we are solving for a length, the solution of –27 must be rejected. The width of the deck is 3 feet.
If a person randomly guesses on each question of a test with n questions, what is the probability of guessing half of the questions correctly if each question has five possible answer choices?
If you are randomly guessing with five possible answer choices, the probability of guessing correct is 1 out of 5, or 1/5. Since the test has n number of questions and we want to get half of them correct, we want this to happen n/2 times. Therefore, the probability would be 1/5 times itself n/2 times, or
Two integers are in the ratio of 1 to 4. If 6 is added to the smaller number, the ratio becomes 1 to 2. Find the larger integer.
Let x = the smaller integer. The ratio of 1 to 4 can be written as 1x to 4x or . Add 6 to the smaller integer, set the ratio equal to 1/2, and solve. . Cross-multiply to get 2x + 12 = 4x. Subtract 2x from both sides of the equation. 2x – 2x + 12 = 4x – 2x. 12 = 2x, so 6 = x. If the smaller integer is 6, then the larger integer is 6 × 4 = 24.
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