Iram J. answered • 08/31/15
Tutor
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Math Tutor Prealgebra to Precalculus
Question #1
One number is 14 more than the other. If the sum of the two numbers is 82, find the two numbers.
One number is 14 more than the other. If the sum of the two numbers is 82, find the two numbers.
This problem will require a system of equations. To solve this, let's substitute "one number" with "x" and "the other" with "y". Now the question will read,
X is 14 more than Y. If the sum of x and y is 82, find the two numbers.
The first sentence becomes x = y + 14. Notice the key word "is" means "=" and "more than" means "+."
The second sentence becomes x + y = 82
x = y + 14
x + y = 82
Now, let's use substitution. How do I know to use substitution and not elimination? Well, technically I can use either, but because the first equation already has "x" isolated on one side of the "=", we can more easily plug in (y + 14) for x in the second equation, as follows:
x = y + 14
x + y = 82
x + y = 82
Now substitute into the second equation which will become:
y + 14 + y = 82 --> [combine like terms, the two y's]
2y + 14 = 82 --> [subtract 14 from both sides]
2y = 68 --> [divide both sides by 2]
y = 34
Now plug in to either equation to find x, I'll pick on the first one.
x = y + 14
x = 34 + 14
x = 48
The two numbers are 48 and 34. You're not done though, until you check your answers by plugging them back in to the original equations we set up.
x = y + 14 → 48 = 34 + 14? Yes!
x + y = 82 → 48 + 34 = 82? Yes!
x + y = 82 → 48 + 34 = 82? Yes!
Question #2
The perimeter of a rectangle is 164 feet. The length is 3 feet less than 4 times the width. Find the dimensions of the rectangle.
The perimeter of a rectangle is 164 feet. The length is 3 feet less than 4 times the width. Find the dimensions of the rectangle.
Perimeter is the distance around an object. For a rectangle it's length + width + length + width. 4 edges. In other words: P = l + w + l + w, so P = 2l + 2w. The problem tells us the perimeter is 164 ft, so 164 = 2l + 2w
They tell us the "l" is 3 ft less than 4 times the "w". Remember that "is" "less than" and "times" correlate to "=", "-", and "x," respectively. So the words can be converted to an equation as so: l = 4w - 3.
Now we have two equations, so we solve either by substitution or elimination. Since the second equation has an isolated "l" on one side of the "=", let's use substitution and plug in the value for "l" into the first equation.
164 = 2l + 2w
l = 4w - 3
Now substitute into the first equation which will become:
164 = 2(4w - 3) + 2w --> [use the distributive property to simplify]
164 = 8w - 6 + 2w --> [combine like terms, the two w's]
164 = 10w - 6 --> [add 6 to both sides]
170 = 10w --> [divide both sides by 10]
164 = 2(4w - 3) + 2w --> [use the distributive property to simplify]
164 = 8w - 6 + 2w --> [combine like terms, the two w's]
164 = 10w - 6 --> [add 6 to both sides]
170 = 10w --> [divide both sides by 10]
17 = w
The question says to find the dimensions, not just the width. So, let's find the length by substituting the w value into either equation. I'll pick on the second one.
l = 4w - 3
l = 4(17) - 3
l = 68 - 3
l = 65
So, your answer is 17 ft by 65 ft. (Check your answers, it will work.)
Question #3
The sum of four consecutive even integers is 748. Find the four even integers.
The sum of four consecutive even integers is 748. Find the four even integers.
What does consecutive mean? It means back-to-back, or in our case, numbers that are in order. What are integers? All negative and positive numbers that are not fractions or decimals, (i.e. -3, -2, -1, 0, 1, 2, 3, etc.) So, one example of 4 consecutive even integers might be 4, 6, 8, and 10. Another might be 18, 20, 22, 24. We need to find such four numbers that add up to 748.
Let's pretend the first number is "x".
x + ___ + ___ + ___ = 748 → Now, the next number is now many units more? Two.
x + (x + 2) + ___ + ___ = 748 → Now, the next number is now many units more than x? Four.
x + (x + 2) + (x + 4) + ___ = 748 → Now, the next number is now many units more than x? Six.
x + (x + 2) + (x + 4) + (x + 6) = 748
I've included parenthesis to show the progression, but for calculation purposes do they matter? No, because we're simply adding. Now let's rearrange to line up the x's and the numbers.
x + x + x + x + 2 + 4 + 6 = 748 --> [combine like terms, the x's and the numbers]
4x + 12 = 748 --> [subtract 12 from both sides]
4x = 736 --> [divide both sides by 4]
x = 184
What are the other three numbers? Well, according to our setup, we said that x was the first number in the range (as we said x + ___ + ___ + ___ = 748). So add 2...three times. 184, 186, 188, and 190. Check to make sure it adds up to 748.
Question #4
Lauren deposits a total of $11,500 in savings account at two different banks. The first bank pays 4.5% interest, while the second bank pays 6% interest. If she earns a total of $637.50 in interest in one year, how much was deposited at each bank?
Lauren deposits a total of $11,500 in savings account at two different banks. The first bank pays 4.5% interest, while the second bank pays 6% interest. If she earns a total of $637.50 in interest in one year, how much was deposited at each bank?
When it comes to word problems, your best bet is to start with the question itself - what is the question asking? How much was deposited in each bank?...so let's define some variables.
a = amount deposited at first bank
b = amount deposited at second bank.
The first sentence says she deposits a total of $11,500 in savings account at two different banks. So this means:
a + b = 11,500
The second and third sentence talk about interest. Interest accumulates on the deposit amount. They tell us the first bank pays 4.5% interest, while the second bank pays 6% interest, and she earns a total of $637.50 in interest in one year. So the interest on amount "a" plus the interest on amount "b" is 637.5...which translates to the equation:
0.045a + 0.06b = 637.5
(where did i get 0.045 and 0.06? Well percentages are numbers out of 100. So if interest is paid at 6%, it's actually the deposit amount times (6/100), which is the decimal 0.06)
For this problem we have a format that allows for elimination, but for simplicity, let's use substitution as we have been. Let's take the first equation, and subtract b from both sides to isolate a. a + b = 11,500 --> a = 11500 - b. Now we can substitute for a into the second equation.
a = 11,500 - b
0.045a + 0.06b = 637.5
0.045(11500 - b) + 0.06b = 637.5 --> [use the distributive property to simplify]
517.5 - 0.045b + 0.06b = 637.5 --> [combine like terms, the two b's]
517.5 + 0.015b = 637.5 --> [subtract 517.5 from both sides]
0.015b = 120 --> [divide both sides by 0.015]
b = 8000
517.5 - 0.045b + 0.06b = 637.5 --> [combine like terms, the two b's]
517.5 + 0.015b = 637.5 --> [subtract 517.5 from both sides]
0.015b = 120 --> [divide both sides by 0.015]
b = 8000
Now, find "a" by using the easier, first equation.
a + b = 11,500
a + 8000 = 11500
a = 3500
Check your answers, and viola!
