GEOMETRIC SEQUENCES AND SUMS

     

Laura received her Master"s degree in Pure Mathematics from Michigan State University, và her Bachelor"s degree in Mathematics from Grand Valley State University. She has trăng tròn years of experience teaching collegiate mathematics at various institutions.

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Dr. Chan has taught computer và college level physics, chemistry, and math for over eight years. Dr. Chan has a Ph.D. In Chemistry from U. C. Berkeley, an M.S. Physics plus 19 graduate Applied Math credits from UW, and an A.B. With honors from U.C .Berkeley in Physics.

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Adding Fractions

In everyday life, we encounter scenarios in which we need to add three fractions. Suppose you want to ride your bike to your friend"s house. Khổng lồ get there, you must first ride 1/2 of a mile down Elm Street, then 1/4 of a mile down Maple Street, và finally 1/8 of a mile down Oak Street. Before you head out, you want lớn know how far you have to ride.

You can showroom the distances on each street, or 1/2 + 1/4 + 1/8. If we"re adding fractions that have the same denominator such as 1/4 và 2/4, all we have to bởi vì is địa chỉ cửa hàng together the numerators to lớn find the result. Since 1 + 2 = 3, our numerator is 3. The denominator, 4, stays the same. Thus, 1/4 + 2/4 = 3/4.

So, how vị we showroom fractions with different denominators? To vì chưng this, we have to add a step khổng lồ our procedure. Before we địa chỉ the numerators, we have lớn convert our fractions so that they have the same denominator.


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0:03 Adding Fractions 1:01 Finding a Common Denominator 2:26 One Example of Adding… 4:59 Another Way to Solve 6:23 Lesson Summary
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Finding a Common Denominator

A common denominator is the same denominator in two or more fractions. The common denominator will be the least common multiple of the denominators of the fractions. The least common multiple (abbreviated LCM) of a group of numbers is the smallest positive number that all of the numbers in the group divide into evenly. To find this, it"s helpful to danh sách multiples of each of the numbers.

Let"s say we want to lớn find a common denominator for the fractions 1/3 and 2/5. We need khổng lồ find the least common multiple of 3 & 5. We can các mục the multiples of each number:

Multiples of 3 are: 3, 6, 9, 12, 15, 18, & so on Multiples of 5 are: 5, 10, 15, 20, & so on

The first number in common in these two lists is 15, so the least common multiple of 3 and 5 is 15.

If we wanted to showroom 1/3 và 2/5, we would first need to lớn convert both fractions lớn denominators of 15. How vì chưng we change the denominator of a fraction without changing the value of that fraction? To do this, we have to lớn multiple both the numerator & denominator by the same number. Lớn change 1/3 into 15ths, we look at what we would need to multiple 3 by khổng lồ get 15. 3 times 5 is 15, so if we multiple the numerator of the fraction (1) by the same number (5) we get 5. 1/3 is equal to 5/15.

One Example of Adding 3 Fractions

Let"s practice these steps lớn find out how far you have lớn ride your bike to get to lớn your friend"s house. We want to add một nửa + 1/4 + 1/8. There are a couple of ways to lớn go about adding three fractions:

The associative property states (a + b) + c = a + (b + c). This means that it doesn"t matter which fractions we showroom together first. We can showroom any two of the fractions together & then địa chỉ the result to the third fraction. Let"s start by adding 50% and 1/4. First we need a common denominator. This will be the least common multiple of the denominators in the fractions. In our case, we want to lớn find the least common multiple of 2 & 4:

Mutliples of 2 are: 2, 4, 6, 8, 10, và so on Multiples of 4 are: 4, 8, 12, 16, and so on

Comparing the list, we see that 4 is the least common multiple of 2 và 4. Next, we need lớn make 4 the denominator of both fractions. We don"t have to vị anything to 1/4 since it already has this denominator. Khổng lồ make 4 the denominator of 1/2, we need lớn multiply both the numerator and denominator by 2, since 2 times 2 equals 4. When we do this, we find that 1/2 is equal lớn 2/4. Now, we"re ready to địa chỉ our fractions. Remember, to showroom two fractions with the same denominator, simply showroom the numerators: the denominator stays the same. When we địa chỉ cửa hàng 2/4 and 1/4, we địa chỉ 2 + 1 = 3 lớn find the numerator. Since the denominator is 4, we find that 2/4 + 1/4 = 3/4.

We"re almost there! Now, we need to địa chỉ our third fraction, 1/8, to lớn the result. We want to lớn find 3/4 + 1/8. Again, we have to lớn first find a common denominator by determining the least common multiple of 4 and 8:


Multiples of 4 are: 4, 8, 12, 16, and so on Multiples of 8 are: 8, 16, 24, & so on

We see that 8 is the least common multiple of 4 and 8. Since 8 is already the denominator for 1/8, we just need to lớn manipulate ba phần tư so that it also has the denominator 8. To bởi vì this, we multiply both the numerator & denominator by 2. This gives us 6/8. Now, we showroom the numerators of 6/8 and 1/8: 6 + 1 = 7, which means that ba phần tư + 1/8 = 7/8. Thus, by first adding 1/2 plus 1/4 and then adding 1/8 lớn the result, we find that 50% + 1/4 + 1/8 = 7/8.

Another Way khổng lồ Solve

Another way khổng lồ solve this equation is to add all three fractions at once. To vị this, we must find a common denominator for all three fractions. In this case, we would start by find the least common multiple of 2, 4, & 8:

Multiples of 2 are: 2, 4, 6, 8, 10, và so on Multiples of 4 are: 4, 8, 12, 16, & so on Multiples of 8 are: 8, 16, 24, và so on

8 is the smallest number each of these numbers divides into evenly. Now, we just need to lớn make 8 our common denominator. The denominator of 1/8 is already 8, so we leave it alone. Khổng lồ get a denominator of 8 for 1/2, we multiple the numerator and denominator by 4, giving us 4/8. Khổng lồ get a denominator of 8 for 1/4, we multiple the numerator & denominator by 2, giving us 2/8. Now we have three fractions with the denominator of 8. We need to địa chỉ 4/8 + 2/8 + 1/8. We simply showroom the numerators: 4 + 2 + 1 = 7. Once again, we find that 1/2 + 1/4 + 1/8 = 7/8.

Both methods yielded the same result, whether we địa chỉ two fractions first & then showroom the last one khổng lồ the result, or địa chỉ cửa hàng all three together at once, the answer is the same. You"d have to ride your bike 7/8 of a mile to get from your house to your friend"s house.

Lesson Summary

When adding fractions, the fractions must have a common denominator. Khổng lồ change fractions so that they have a common denominator, we first find the least common multiple, or the smallest positive number that all of the numbers in the group divide into evenly. We find this by making a các mục of the multiples of each denominator. Once we"ve found a common denominator, we convert our fractions by multiplying both the numerator và denominator by the same number, so that the fraction does not change its value. Then, we simply địa chỉ together the numerators and keep the denominator the same to lớn give us our result. Because of the associative property, which states that numbers can be added in any order, it does not matter which fractions we showroom together first. We can also showroom all three at once. Knowing how to địa chỉ cửa hàng fractions is very useful for real-world applications.


Practice Adding Fractions

These questions will give you practice adding fractions. Answer the questions, và then kiểm tra your work using the answers in the Answer Key.

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Question 1

What is 1/2 + 1/4 + 1/8 equal khổng lồ (in lowest terms)?

a) 7/8

b) 7/4

c) 5/8

d) 7/16

Question 2

What is một nửa + 1/4 + 1/8 + 1/7 in lowest terms?

a) 57/56

b) 57/58

c) 58/59

d) 56/57

Answer Key

To địa chỉ fractions like một nửa + 1/4 + 1/8, you take the largest denominator (D) and try to lớn convert every fraction to an equivalent fraction with that denominator. Before doing that, it might be possible khổng lồ reduce some fractions khổng lồ smaller denominators if the denominator can be divided by the numerator.

Answer 1

a) 7/8

Find largest denominator:

D=2(4)(8)= 64

Convert khổng lồ equivalent fractions, add, & reduce to lớn lowest terms:

1/2 + 1/4 + 1/8 = 32/64 + 16/64 + 8/64 = (32 + 16 + 8) / 64 = 56/64 = 7/8

Answer 2

a) 57/56

Find the largest denominator:

D=2(4)(8)7 = 64(7) = 448

Convert to lớn equivalent fractions:

1/2 + 1/4 + 1/8 + 1/7 = (1/2)448/448 + (1/4)448/448 + (1/8) 448/448 + (1/7) 448/448 = 224/448 + 112/448 + 56/448 + 64/ 448