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Paraphin [41]
2 years ago
6

How many different integers between $100$ and $500$ are multiples of either $6,$ $8,$ or both?

Mathematics
1 answer:
nirvana33 [79]2 years ago
8 0
We need to find the number of integers between 100 and 500 that can be divided by 6, 8, or both. Now, to do this, we must as to how many are divisible by 6 and how many are multiples of 8.

The closest number to 100 that is divisible by 6 is 102. 498 is the multiple of 6 closest to 500. To find the number of multiple of 6 from 102 to 498, we have

n = \frac{498-102}{6} + 1
n = 67

We can use the same approach, to find the number of integers that are divisible by 8 between 100 and 500. 

n = \frac{496-104}{8} + 1
n = 50

That means there are 67 integers that are divisible by 6 and 50 integers divisible by 8. Remember that 6 and 8 share a common multiple of 24. That means the numbers 24,  48, 72, 96, etc are included in both lists. As shown below, there are 16 numbers that are multiples of 24.

n = \frac{480-120}{24} + 1
n = 16

Since we counted them twice, we subtract the number of integers that are divisible by 24 and have a final total of 67 + 50 - 16 = 101. Hence there are 101 integers that are divisible by 6, 8, or both.

Answer: 101


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Mel wants to buy a pair of shoes that costs $120. She has saved $45 and has a job that pays her $10 per hour. The number of hour
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<h2>Answer: 8</h2>

Step-by-step explanation:

10x+45 > 120\\10x+45-45 > 120-45\\10x > 75\\\frac{10x}{10} > \frac{75}{10} \\x > 7.5

Therefore, Mel has to work at least 7.5 hours or 8 total hours.

3 0
1 year ago
John, Sally, and Natalie would all like to save some money. John decides that it would be best to save money in a jar in his clo
Radda [10]

Answer:

Part 1) John’s situation is modeled by a linear equation (see the explanation)

Part 2) y=100x+300

Part 3) \$12,300

Part 4) Is a exponential growth function

Part 5) A=6,000(1.07)^{t}  

Part 6) \$11,802.91  

Part 7) Is a exponential growth function

Part 8) A=5,000(e)^{0.10t}    or  A=5,000(1.1052)^{t}  

Part 9)  \$13,591.41

Part 10) Natalie has the most money after 10 years

Step-by-step explanation:

Part 1) What type of equation models John’s situation?

Let

y ----> the total money saved in a jar

x ---> the time in months

The linear equation in slope intercept form

y=mx+b

The slope is equal to

m=\$100\ per\ month

The y-intercept or initial value is

b=\$300

so

y=100x+300

therefore

John’s situation is modeled by a linear equation

Part 2) Write the model equation for John’s situation

y=100x+300

see part 1)

Part 3) How much money will John have after 10 years?

Remember that

1 year is equal to 12 months

so

10 years=10(12)=120 months

For x=120 months

substitute in the linear equation

y=100(120)+300=\$12,300

Part 4) What type of exponential model is Sally’s situation?

we know that    

The compound interest formula is equal to  

A=P(1+\frac{r}{n})^{nt}  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest  in decimal

t is Number of Time Periods  

n is the number of times interest is compounded per year

in this problem we have  

P=\$6,000\\ r=7\%=0.07\\n=1  

substitute in the formula above

A=6,000(1+\frac{0.07}{1})^{1*t}  

A=6,000(1.07)^{t}  

therefore

Is a exponential growth function

Part 5) Write the model equation for Sally’s situation

A=6,000(1.07)^{t}  

see the Part 4)

Part 6) How much money will Sally have after 10 years?

For t=10 years

substitute  the value of t in the exponential growth function

A=6,000(1.07)^{10}=\$11,802.91  

Part 7) What type of exponential model is Natalie’s situation?

we know that

The formula to calculate continuously compounded interest is equal to

A=P(e)^{rt}  

where  

A is the Final Investment Value  

P is the Principal amount of money to be invested  

r is the rate of interest in decimal  

t is Number of Time Periods  

e is the mathematical constant number

we have  

P=\$5,000\\r=10\%=0.10  

substitute in the formula above

A=5,000(e)^{0.10t}  

Applying property of exponents

A=5,000(1.1052)^{t}  

therefore

Is a exponential growth function

Part 8) Write the model equation for Natalie’s situation

A=5,000(e)^{0.10t}    or  A=5,000(1.1052)^{t}

see Part 7)

Part 9) How much money will Natalie have after 10 years?

For t=10 years

substitute

A=5,000(e)^{0.10*10}=\$13,591.41

Part 10) Who will have the most money after 10 years?

Compare the final investment after 10 years of John, Sally, and Natalie

Natalie has the most money after 10 years

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Answer:

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