Middle and High School Math
Problem 1: Leaves from a tree were reported by four different European students to be 2.9 cm, 3.33 cm, 3.9 cm, and 3.12 cm in length. List the numbers in order of decreasing length.
For the beginner, the easiest way to evaluate which of these number is smaller and which is larger, is to make the number of digits to the right of the decimal the same. Now the maximum number of such digits here is two.
Adding zeros to the right of the last digit does not change a number’s value. When that is done, the numbers become:
2.90, 3.33, 3.90 and 3.12
In decreasing order, the lengths are:
3.90 cm, 3.33 cm, 3.12 cm and 2.90 cm. [Answer]
Problem 2: Find the midpoint of the segment from Point A (2, –7) to Point B (8, –1).
Given the two endpoints, the midpoint is derived by finding the halfway value between the two first values (in this instance, the 2 and the 8, hence 5) and the halfway value between the two second values (in this instance, the –7 and the –1, hence – 4).
The midpoint of the segment is:
(5, – 4) [Answer]
Problem 3: 5/12–y+5/4 =2/3. What is y?
There are no parentheses in this problem. Thus the first two terms are simply 5/12 – y, rather than 5/(12-y). So we write the problem (including spaces for clarity),
5/12 – y + 5/4 = 2/3
Multiplying this equation by –1 changes all the signs, producing,
– 5/12 + y – 5/4 = – 2/3
Now, when the same number is added on both sides of the equal sign, nothing is actually changed. So we add + 5/12 and + 5/4 to both sides to give,
y = –2/3 + 5/12 + 5/4
We will change all the fractions into 12ths. Numbers like 3/3 and 4/4 are equal to 1, so multiplying a fraction by them does not change the value of the fraction. We get,
y = –2/3 (4/4) + 5/12 (1/1) + 5/4 (3/3) = –8/12 + 5/12 + 15/12 = 12/12 = 1
y = 1 [Answer]
Problem 4: Given these two equations,
3x + 2y = A
5x + y = B
Solve for x in terms of A and B.
The first equation by simple algebraic manipulation (moving the 3x to the right and dividing by 2) becomes,
y = (A – 3x)/2
For the second equation, move the 5x to the right to obtain,
y = B – 5x
Combining our two modified equations gives us,
(A – 3x) / 2 = B – 5x
Multiplying both sides by 2 gives,
A – 3x = 2B – 10x
This simplifies (by subtracting – 3x from both sides of the equal sign) to,
A = 2B – 7x
So by simply moving terms and dividing, we obtain,
x = (2B – A) / 7 [Answer]
Problem 5: What is 3 divided by 10, shown as a fraction?
Most simply, it can be written as 3/10, with the slash “/” meaning “divide by”. But there is another way to write the answer, using decimals.
Division by ten calls for a move in decimal point one place to the left – one space. Consider:
3 may be written 3.0
Dividing it by 10 makes it
Because the decimal point is sometimes missed, a fraction less than 1 is customarily written with a zero in front of the decimal. Thus we would write .30 as,
Problem 6: What is the answer to this problem? (8/2)/2+((12+3)+25)=?
This problem is obviously intended to demonstrate which mathematical operations are carried out first, which are carried out second, and so forth. It matters, because done out of sequence, a wrong answer is likely to be obtained.
The first operations that should be carried out here are those operations within parentheses. In fact, the very first operation is the parentheses within parentheses! Within parentheses, multiplication and division are carried out first, and then addition and subtraction. We begin,
(4)/2 + ((15) + 25)
So we wind up with
2 + 40 = 42 [Answer]
Problem 7: A farmer wants to put in a fence with five sides around his field. The measurements are 50 ft, 12 ft, 60 ft, 35 ft, and 25 ft. Find the perimeter in feet and convert the number to yards.
The perimeter, which we’ll call P, is the sum of the sides,
P = 50 ft + 12 ft + 60 ft + 35 ft + 25 ft = 182 feet
Now, 1 yd = 3 ft So,
182 ft x 1 yd / 3 ft = 182 ft / 3 yd = 60-2/3 yd [Answer]
Problem 8: The midpoint of the line segment from P1 to P2 is (–3, 2) if P1 = (–5, 2) what is P2?
The midpoint of a line segment has values that are intermediate between both the first (usually x) variable value of the points as well as the second (usually y) variable value of the points.
The y values are the same for the midpoint and one endpoint. This means the line is horizontal and so the other end point has the same value, 2.
The x values vary. One endpoint’s x-value is –5. The midpoint has x-value –3. The difference between the one end and the midpoint is 2 units, so the difference between the midpoint and the other endpoint is 2 units. Hence, the other endpoint has value –1.
(–1, 2) [Answer]