Math Problems Plus SolutionsProblem 1: Find the slope intercept form of the line passing through the point (– 1, 5) and parallel to the line – 6x – 7y = – 3.
The line given is rewritten (in slope-intercept form, y = mx + b) as
y = – 6/7 x + 3/7Thus the slope m = – 6/7
Now two lines are parallel if they have the same slope. So,
y = – 6/7 x + bis the formula for the new line, with the intercept not yet solved. We do so by inserting the value of x and the value of y from the point given (– 1, 5). We write,
5 = (– 6/7)(– 1) + bRearranging and performing the arithmetic,
b = 29/7The parallel line is represented by the equation:
y = – 6/7x + 29/7 [Answer]————————-
Problem 2: There is a rectangular garden that measures 20 x 30 feet. You want to double the area of the garden. What would be the length and width if one side is 1.50 times greater than the other. What would their measure be to the nearest 10th of a foot?
the area of a rectangle = length x width
Let’s assign to the length the variable name x and to the width the variable name y.
So the area of the garden is,
area = 20 x 30 = 600Recall that x = 1.5 y. This is the same (for the purposes of this discussion) as x = 3/2 y. So, replacing x with 3/2 y we get,
area = (3/2 y)(y) = 3/2 y²Now the new garden will have double the area or 1,200 square feet.
We write our new equation as,
1,200 = 3/2 y² or its equivalent, 3/2 y² = 1,200Multiplying both sides of the equation and taking the positive square root of y² we get,
y = √800 or 28.283 feet, or, to the nearest tenth of a foot, 28.3.Now recall that
x = 1.5 ySo x = 1.5 (28.283) = 42.424 feet or, to the nearest tenth of a foot, 42.4.
x = 42.4, y = 28.3 [Answer]Does this check?
42.4 x 28.3 = 1199.9 square feet. [Check Correct]————————-
Problem 3: If 2 inches corresponds to 125 yards, what is the scale of a map? The scale of a map is a pure ratio, and is generally written something like 1:4, 1:50 or 1:17,000. The ratio is the same whether the unit is a millimeter, a furlong, or a light-year. It doesn’t matter. The goal is to make the units the same. We convert to inches. If 2 inches corresponds to 125 yards, it also corresponds to 4,500 inches, since 125 yards = 4,500 inches. Is the scale 1:4,500 then? No, because 4,500 corresponds to 2 inches, not 1 inch. The scale is,
Problem 4: y = 1/2x – 10 How do I write this in standard form using integers?
A fraction is not an integer. Integers are whole numbers. The only fraction in our expression – as written here – is the 1/2. So we multiply both sides of the equation by 2 on both sides, to get,
(2) y = (2)(1/2x – 10)So,
2y = x – 20, or
x – 2 y = 20 [Answer]————————-
Problem 5: Problem: If F(x) = x² – 3x + 4 , determine f(x + h) writing it in the simplified way.
This factorable quadratic equation can be rewritten in the form,
f(x) = (x – 4)(x + 1)
f(x + h) = ((x + h) – 4)((x + h) + 1)But this is the same as,
f(x + h) = (x + h – 4)(x + h + 1)
f(x + h) = x² + 2xh – 3x – 3h – 4 [Answer]————————-
Problem 6: An increase in temperature of 10 degrees in Celsius represents how much of an increase in Fahrenheit?
Let’s consider four points, two for Fahrenheit, two for Celsius. We know water freezes at 32 degrees F and boils at 212 degrees F. In degrees Celsius, water freezes at 0 and boils at 100.
The degree differences between freezing and boiling are 180 for F and 100 for C. This means that a degree Fahrenheit is 5/9 the measure of a degree Celsius.
So if we increase temperature by 10 degrees Celsius, we have increased the Fahrenheit temperature by 9/5 x 10 = 18 degrees.
18 degrees [Answer]————————-
Problem 7: If I earn $18 an hour by reading 100 meters, how many meters must I read in an hour to earn $25?
This can be determined by setting up in this format:
x meters/$25 = 100 meters/$18For simplicity, we drop the $ and meters.
The equation now reads,
x/25 = 100/18Multiply both sides by 25. This changes nothing. Consider: everybody knows 2 = 2. If we multiply both sides by 25, we get 50 = 50. Still true.
Returning to our problem: multiplying both sides by 25, we get,
x = 2500/18
x = 138.88 meters.Of course, we cannot read a fractional meter.
139 meters [Answer]————————-
Problem 8: The difference of triple a number plus 1/2 is equal to the sum of the number plus 2/3. Calculate the number.
Let the number be represented by the variable x.
3x + 1/2 = x + 2/3Move all x to the left and the pure numbers to the right. We now have,
3x – x = 2/3 – 1/2Now 2/3 is the same as 4/6 and – 1/2 is the same as – 3/6. So this simplifies to,
2x = 1/6.
x = 1/12 [Answer]Let’s perform the check. We get,
3(1/12) + 1/2 = 1/12 + 2/3
3/12 + 6/12 = 1/12 + 8/12
9/12 = 9/12 [Check Correct]It is hoped you enjoyed these math problems plus solutions. Please feel free to suggest different problems in the comments section, below.
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