Answers to Problems of the Week - March 8 to March 13

Rewards for Problems of the week:

5 points for the correct answer at your level; 10 points for the next level; 20 points each for 2+ levels up.

 Lower Elementary:

Question: What are the next 4 numbers in the pattern?

50, 45, 46, 41, 42, _____, _____, _____, _____...

Answer:  37, 38, 33, 34

Solution:  There are two steps in the pattern. To get from 50 to 45, we subtract 5. To get from 45 to 46, we add 1. Then the pattern repeats: 46 – 5 = 41, 41 + 1 = 42, 42 – 5 = 37, 37 + 1 = 38, 38 – 5 = 33, 33 + 1 = 34.

 Upper Elementary:

Question: Chloe gets on her bike at home and rides 2 miles north to the post office. From there, she rides 3 miles west to the library. Then, she rides 4 miles south to the park, then 5 miles east to the farmer’s market. Finally, Chloe rides her bike 2 miles north to her friend Daniel’s house. How far away is Daniel’s house from Chloe’s house? In what direction?

Answer:  2 miles east

Solution:  Instead of solving this mentally, students can use a coordinate plane and put Chloe’s house on the origin. The post office is at (0, 2). The library is at (^–3, 2). The park is at (^–3, ^–2). The farmer’s market is at (2, ^–2). Finally, Daniel’s house is at (2, 0), which is 2 miles to the right, or east, of Chloe’s house.

Middle School:

Question: The figure below is horizontally and vertically symmetrical. What is its area?

(The figure is not drawn to scale.)

Answer:  104 square centimeters

Solution:  One way we can find the area of the shape by thinking of it as two identical trapezoids. The area of a trapezoid is the average of its two bases multiplied by its height. The bases are 16 cm and 10 cm, so the average is 13 cm. The height is 4 cm. So, the area of one of the trapezoids is 13 × 4 = 52 square centimeters. The area of the whole figure is therefore 52 × 2 = 104 square centimeters.

 Algebra and Up:                                                                             

Question: There are two perpendicular lines graphed on a plane. Line A goes through the points (2, 1) and (6, 3). Line B goes through (2, 1) and (x, 5). Solve for x.

Answer:  x = 0

Solution:  To solve for x, we need to know the slope of Line B. Line A and Line B are perpendicular, so the slope of Line B is the opposite of the reciprocal of the slope of Line A. Line A’s slope is ½, so the slope of Line B must be ^–2. Now that we have a point on the line and its slope, we can write an equation in slope-intercept form:

y = ^–2x + b

y = 1   x = 2

1 = ^–4 + b

b = 5

y = ^–2x + 5

5 is the y–intercept. When y = 5, then x = 0.

Challenge problem to take home. 20 points for the correct answer. 

$\\[1.7em]$

How many triangles would balance three circles?

Answer: 2 Triangles