Adding and subtracting fractions with unlike denominators

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Why fractions need a common denominator

You can only add or subtract fractions when the denominators (the bottom numbers) are the same.
The denominator tells you the size of the pieces. If pieces are different sizes, you can't count them together.
Think of 1/2 + 1/3: halves and thirds are different sizes, so you can't just add the top numbers.
To add or subtract, first rewrite the fractions so they have the same denominator.

Equivalent fractions

An equivalent fraction is a fraction that has the same value but a different top and bottom number.
You can make an equivalent fraction by multiplying or dividing the top and bottom by the same number.
For example, 1/2 = 2/4 = 3/6 because you multiply top and bottom by 2 or 3.
Multiplying by a fraction like 2/2 is the same as multiplying by 1, so the value doesn't change.

Finding a common denominator

A common denominator is a number that is a multiple of both denominators.
The easiest common denominator is to multiply the two denominators together.
For 1/2 + 1/3, multiply 2 and 3 to get 6 as a common denominator.
But sometimes a smaller common denominator exists, called the lowest common denominator (LCD).

Lowest common denominator (LCD)

The lowest common denominator (LCD) is the smallest number that is a multiple of both denominators.
It is also called the least common multiple of the denominators.
For 5/12 + 11/18, the LCD is 36 because 36 is the smallest number divisible by both 12 and 18.
Using the LCD keeps the numbers smaller and makes the answer easier to simplify.

Steps to add or subtract fractions with unlike denominators

Step 1: Find a common denominator (preferably the LCD).
Step 2: Rewrite each fraction as an equivalent fraction with that denominator.
Step 3: Add or subtract the numerators (top numbers) and keep the denominator the same.
Step 4: Simplify the answer if possible.

Adding fractions with unlike denominators

Example: 1/2 + 2/3.
The LCD of 2 and 3 is 6.
Rewrite: 1/2 = 3/6 and 2/3 = 4/6.
Add: 3/6 + 4/6 = 7/6.
7/6 is an improper fraction (top bigger than bottom). You can leave it or change it to 1 1/6.

Subtracting fractions with unlike denominators

Example: 5/6 - 1/4.
The LCD of 6 and 4 is 12.
Rewrite: 5/6 = 10/12 and 1/4 = 3/12.
Subtract: 10/12 - 3/12 = 7/12.
7/12 is already in simplest form.

Mixed numbers

A mixed number has a whole number and a fraction, like 2 1/3.
To add or subtract mixed numbers, you can work with the whole parts and fraction parts separately.
Sometimes you need to borrow (exchange) a whole for a fraction when subtracting.
Example: 3 1/4 - 1 3/4 – you can't subtract 3/4 from 1/4, so borrow 1 from 3 to make it 2 5/4.
Then subtract: 2 5/4 - 1 3/4 = 1 2/4 = 1 1/2.

Simplifying answers

Always check if your answer can be simplified.
Simplify by dividing the top and bottom by the greatest common factor (GCF).
For example, 2/4 simplifies to 1/2 because you divide top and bottom by 2.
If the answer is an improper fraction, you can change it to a mixed number.

Estimating to check answers

Use benchmark fractions like 0, 1/2, and 1 to estimate.
For 1/2 + 2/3, estimate: 1/2 is about 0.5 and 2/3 is about 0.67, so the sum is about 1.17.
Your exact answer 7/6 is about 1.17, so it makes sense.
Estimating helps you catch big mistakes.

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練習問題

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1.What is the lowest common denominator (LCD) of the fractions 1/4 and 1/6?
Easy
- A12
- B24
- C4
- D6
2.Which fraction is equivalent to 2/3?
Easy
- A6/9
- B4/9
- C5/6
- D3/4
3.You can add fractions with different denominators without changing them first.
Easy
True or false?
4.Add: 1/2 + 2/3 = ?
Medium
- A7/6
- B3/5
- C5/6
- D1/6
5.What is the least common multiple of 3 and 5?
Medium
- A15
- B8
- C30
- D10
6.Match each sum to its correct answer.
Medium
- 1/4 + 1/3
- 1/5 + 1/2
- 2/3 + 1/6
- 7/12
- 7/10
- 5/6
7.Order these fractions from smallest to largest: 1/2, 3/4, 2/3
Medium
- 1/2
- 2/3
- 3/4
8.Which of the following are equivalent to 2/3? (select all that apply)
Hard
- A4/6
- B6/9
- C8/12
- D5/9
- E10/15

もっと問題

9.Match each pair of fractions to the lowest common denominator (LCD) you would use to add them.
Medium
- 1/3 and 1/4
- 1/6 and 1/8
- 2/5 and 3/10
- 12
- 24
- 10
10.Put these steps in the correct order to add 1/4 and 1/6.
Medium
- Rewrite 1/4 as 3/12 and 1/6 as 2/12.
- Find the lowest common denominator, which is 12.
- Add the numerators: 3 + 2 = 5.
- Write the answer as 5/12.
11.Which of the following are correct ways to add 1/2 and 1/3? (select all that apply)
Medium
- A1/2 + 1/3 = 3/6 + 2/6 = 5/6
- B1/2 + 1/3 = 2/5
- C1/2 + 1/3 = 2/6 + 3/6 = 5/6
- D1/2 + 1/3 = 5/6
- E1/2 + 1/3 = 1/5
12.Which is the lowest common denominator (LCD) for the fractions 2/3 and 1/5?
Easy
- A8
- B15
- C3
- D5
13.To subtract fractions with different denominators, you must first rewrite them with a common denominator.
Medium
True or false?
14.Match each expression to its simplified answer.
Medium
- 5/6 + 1/2
- 7/10 - 1/5
- 2/3 + 1/4
- 4/3
- 1/2
- 11/12
15.Which of the following are true about the sum 1/4 + 1/3? (select all that apply)
Medium
- AThe LCD is 12.
- BThe sum is 2/7.
- CThe sum is 7/12.
- DYou can rewrite 1/4 as 3/12.
- EYou can rewrite 1/3 as 4/12.
16.Arrange these steps to subtract 3/5 - 1/2.
Medium
- Rewrite 3/5 as 6/10 and 1/2 as 5/10.
- Find the lowest common denominator, which is 10.
- Subtract the numerators: 6 - 5 = 1.
- Write the answer as 1/10.
17.Which is the lowest common denominator (LCD) for the fractions 1/3 and 1/4?
Easy
- A7
- B12
- C24
- D3
18.What is 1/3 + 1/4?
Medium
- A2/7
- B7/12
- C2/12
- D1/12
19.To add fractions with different denominators, you must first rewrite them with a common denominator.
Easy
True or false?
20.What is 3/4 + 5/6?
Medium
- A8/10
- B19/12
- C8/24
- D1 2/12
21.Which of the following are correct steps to add 1/2 and 1/5? (select all that apply)
Medium
- AFind the LCD, which is 10.
- BRewrite 1/2 as 5/10 and 1/5 as 2/10.
- CAdd the numerators: 5 + 2 = 7.
- DKeep the denominator as 10.
- EThe answer is 7/20.
22.Match each fraction to its equivalent fraction with denominator 12.
Medium
- 1/3
- 1/4
- 5/6
- 4/12
- 3/12
- 10/12
23.Put these steps in the correct order to subtract 3/4 - 1/3.
Medium
- Find the LCD of 4 and 3.
- Rewrite 3/4 as 9/12 and 1/3 as 4/12.
- Subtract the numerators: 9 - 4 = 5.
- Write the answer over the common denominator: 5/12.
24.A piece of ribbon is 7/8 meter long. You cut off 1/3 meter. How much ribbon is left?
Medium
- A5/24 meter
- B13/24 meter
- C19/24 meter
- D1/2 meter
25.A rectangular garden has a length of 3/4 meter and a width of 2/5 meter. What is the perimeter of the garden?
Hard
- A1 3/10 meters
- B2 3/10 meters
- C1 1/2 meters
- D2 1/2 meters
26.A pizza has 3/4 of a pepperoni pizza left and 2/3 of a cheese pizza left. How much pizza is left in total?
Medium
- A1 5/12 pizzas
- B1 1/12 pizzas
- C5/7 pizza
- D1 1/2 pizzas
27.A rectangular picture frame has an area of 3/5 square meter. The width is 2/3 meter. What is the length?
Hard
- A9/10 meter
- B2/5 meter
- C1 1/5 meters
- D4/5 meter
28.Which of the following are true about adding 1/2 and 1/3? (select all that apply)
Medium
- AYou can add them directly because they have the same denominator.
- BThe lowest common denominator is 6.
- C1/2 + 1/3 = 5/6
- DYou must rewrite both fractions with a common denominator before adding.
- E1/2 + 1/3 = 2/5
29.To subtract 3/4 - 1/6, you can use 12 as a common denominator.
Medium
True or false?
30.Arrange these steps to add 2/3 and 1/4.
Medium
- Find the LCD of 3 and 4.
- Rewrite 2/3 as 8/12.
- Rewrite 1/4 as 3/12.
- Add the numerators: 8 + 3 = 11.
31.What does LCD stand for in fractions?
Easy
- ALowest common denominator
- BLargest common divisor
- CLeast common multiple
- DLong common division
32.Why must you rewrite fractions with a common denominator before adding or subtracting them?
Easy
- ABecause the numerators must be the same
- BBecause you can only add or subtract fractions with the same denominator
- CBecause the denominators must be different
- DBecause the fractions must be improper
33.The lowest common denominator of two fractions is always the product of their denominators.
Easy
True or false?
34.Which of the following is the lowest common denominator (LCD) for the fractions 5/12 and 11/18?
Easy
- A216
- B36
- C72
- D108
35.Place the following fractions in order from smallest to largest: 2/3, 5/6, 3/4
Medium
- 2/3
- 3/4
- 5/6
36.The product of the denominators is always the lowest common denominator.
Easy
True or false?
37.Match each pair of fractions to the correct common denominator you would use to add them.
Easy
- 1/2 and 1/3
- 3/4 and 5/6
- 2/5 and 3/10
- 6
- 12
- 10
38.Which of the following are true about subtracting 3/4 - 1/6? (select all that apply)
Medium
- AThe common denominator is 12
- B3/4 can be rewritten as 9/12
- C1/6 can be rewritten as 2/12
- DThe answer is 7/12
- EThe answer is 8/12
39.To add or subtract fractions, they must have the same denominator.
Easy
True or false?
40.What is 5/12 + 11/18? Give your answer as an improper fraction in simplest form.
Medium
- A37/36
- B16/30
- C216/216
- D222/216
41.What is the lowest common multiple of 6 and 8?
Medium
42.Add: 5/12 + 11/18 = ? Give your answer as an improper fraction in simplest form.
Medium
43.Fill in the missing numerator to make an equivalent fraction: 2/3 = ?/9
Easy
2/3 = ____/9
44.What is the first step to add 1/4 and 1/6?
Medium
45.Complete the sentence.
Medium
When adding fractions with different denominators, you must first find a ____ denominator.
46.What is the first thing you must do before adding 1/4 and 1/6?
Medium
47.What is the least common multiple (LCM) of 4 and 6?
Medium
48.Complete the equivalent fraction.
Easy
1/4 = ____ /12
49.What is the first step to add 1/6 and 2/5?
Easy
50.Fill in the missing number to make an equivalent fraction.
Medium
3/4 = ____/12
51.Why can't you add 1/2 and 1/3 directly without changing them?
Easy
52.A recipe needs 2/3 cup of sugar and 1/4 cup of brown sugar. How many cups of sugar are needed in total?
Medium
53.Explain why you cannot add 1/2 and 1/3 directly without changing them.
Hard
54.Explain why you cannot add 1/3 and 1/4 directly without changing them.
Medium
55.What is the lowest common multiple of 6 and 8?
Medium
- A26
- B28
- C24
- D22
56.Add: 5/12 + 11/18 = ? Give your answer as an improper fraction in simplest form.
Medium
- A33
- B41
- C45
- D37
57.What is the least common multiple (LCM) of 4 and 6?
Medium
- A12
- B11
- C13
- D14
58.A recipe needs 2/3 cup of sugar and 1/4 cup of brown sugar. How many cups of sugar are needed in total?
Medium
- A2.9167 cups
- B1.9167 cups
- C-0.0833 cups
- D0.9166666666666666 cups

復習ノート

## Adding and Subtracting Fractions with Unlike Denominators ### Why do we need a common denominator? You can only add or subtract fractions when they have the **same denominator** (the bottom number). If the denominators are different, you must first rewrite the fractions so they share a **common denominator**. ### What is a common denominator? A common denominator is a number that is a multiple of both denominators. The **lowest common denominator (LCD)** is the smallest such number. Using the LCD keeps the numbers small and easy to work with. ### How to find the LCD 1. List the multiples of each denominator. 2. Find the smallest multiple that appears in both lists. **Example:** For 1/2 and 2/3: - Multiples of 2: 2, 4, 6, 8, ... - Multiples of 3: 3, 6, 9, ... - LCD = 6 ### Rewriting fractions with the LCD Multiply the top and bottom of each fraction by the same number so the denominator becomes the LCD. This does not change the value of the fraction. **Example:** \(\frac{1}{2} + \frac{2}{3}\) - \(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\) - \(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\) - Now add: \(\frac{3}{6} + \frac{4}{6} = \frac{7}{6}\) ### Adding and subtracting fractions Once the denominators are the same, just add or subtract the numerators (top numbers) and keep the denominator the same. Then simplify if possible. **Example:** \(\frac{5}{12} + \frac{11}{18}\) - LCD of 12 and 18 is 36. - \(\frac{5}{12} = \frac{15}{36}\), \(\frac{11}{18} = \frac{22}{36}\) - Sum: \(\frac{15}{36} + \frac{22}{36} = \frac{37}{36}\) ### Working with mixed numbers A **mixed number** has a whole number and a fraction, like 2 1/3. To add or subtract mixed numbers, you can either: - Convert them to **improper fractions** (where the numerator is bigger than the denominator), or - Work with the whole parts and fraction parts separately. When subtracting, you may need to **exchange** (borrow) a whole from the whole number part to make the fraction larger. **Example:** \(3 \frac{1}{4} - 1 \frac{3}{4}\) - Convert to improper: \(3 \frac{1}{4} = \frac{13}{4}\), \(1 \frac{3}{4} = \frac{7}{4}\) - Subtract: \(\frac{13}{4} - \frac{7}{4} = \frac{6}{4} = 1 \frac{1}{2}\) ### Simplifying answers Always check if your answer can be simplified. Divide the numerator and denominator by their greatest common factor (GCF). **Example:** \(\frac{6}{4}\) simplifies to \(\frac{3}{2}\) because both 6 and 4 can be divided by 2. ### Estimating to check Use **benchmark fractions** like 0, 1/2, and 1 to see if your answer makes sense. For example, 1/2 + 2/3 is a little more than 1, so 7/6 (which is 1 1/6) is sensible. ### Key idea Always find a common denominator before adding or subtracting fractions. The LCD makes the math easier.

denominator: The bottom number in a fraction. It shows how many equal parts the whole is divided into.
numerator: The top number in a fraction. It shows how many parts you have.
common denominator: A number that is a multiple of the denominators of two or more fractions. It lets you add or subtract them.
lowest common denominator (LCD): The smallest number that is a multiple of all the denominators. Using it keeps numbers small.
equivalent fraction: A fraction that has the same value as another fraction but uses different numbers, like 1/2 and 2/4.
mixed number: A number that has a whole part and a fraction part, like 2 1/3.
improper fraction: A fraction where the numerator is bigger than or equal to the denominator, like 7/4.
simplify: To make a fraction smaller by dividing the top and bottom by the same number, without changing its value.
benchmark fraction: A common fraction used for estimating, like 0, 1/2, and 1.

フラッシュカード

What must two fractions have before you can add or subtract them?

They must have the same denominator (bottom number).

What is a common denominator?

A number that is a multiple of both denominators. For example, 6 is a common denominator for 1/2 and 1/3.

What is the lowest common denominator (LCD)?

The smallest number that is a multiple of both denominators. For 1/2 and 1/3, the LCD is 6.

How do you find the LCD of two fractions?

List the multiples of each denominator and find the smallest one they share. For 2 and 3, multiples of 2: 2,4,6,8...; multiples of 3: 3,6,9...; the smallest shared is 6.

What is an equivalent fraction?

A fraction that has the same value as another but with a different numerator and denominator. For example, 1/2 = 2/4 = 3/6.

How do you rewrite a fraction with a new denominator?

Multiply the numerator and denominator by the same number. For example, to change 1/2 to sixths, multiply top and bottom by 3: 1/2 = 3/6.

Add: 1/2 + 1/3

First find the LCD: 6. Rewrite: 1/2 = 3/6, 1/3 = 2/6. Then add: 3/6 + 2/6 = 5/6.

Subtract: 3/4 - 1/6

LCD is 12. Rewrite: 3/4 = 9/12, 1/6 = 2/12. Subtract: 9/12 - 2/12 = 7/12.

What is an improper fraction?

A fraction where the numerator (top) is greater than or equal to the denominator (bottom). For example, 7/6.

What is a mixed number?

A whole number plus a proper fraction. For example, 1 1/6 means 1 + 1/6.

How do you change an improper fraction to a mixed number?

Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same. For 7/6, 7 ÷ 6 = 1 remainder 1, so 1 1/6.

How do you change a mixed number to an improper fraction?

Multiply the whole number by the denominator, add the numerator, and put that over the denominator. For 1 1/6: (1×6)+1=7, so 7/6.

Add: 1 1/2 + 2 1/3

Add whole numbers: 1+2=3. Add fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. So the answer is 3 5/6.

Subtract: 3 1/4 - 1 2/3

Subtract fractions: 1/4 - 2/3 = 3/12 - 8/12 = -5/12. Since negative, borrow 1 from 3 (making it 2) and add 12/12 to the fraction: 12/12 + 3/12 = 15/12. Then 15/12 - 8/12 = 7/12. So 2 7/12.

What does it mean to simplify a fraction?

To write it in lowest terms by dividing the numerator and denominator by their greatest common factor. For example, 4/8 simplifies to 1/2.

Simplify: 6/8

Divide top and bottom by 2: 6/8 = 3/4.

Why do we need a common denominator to add fractions?

Because the denominator tells the size of the parts. You can only add parts that are the same size. For example, 1/2 + 1/3 is not 2/5 because halves and thirds are different sizes.

What is a benchmark fraction?

A common fraction used for estimating, like 0, 1/2, and 1. For example, 3/8 is close to 1/2.

Estimate: Is 3/4 + 1/8 more or less than 1?

3/4 is close to 1, and 1/8 is small, so the sum is a little more than 1. Actually, 3/4 + 1/8 = 6/8 + 1/8 = 7/8, which is less than 1. So the estimate should be less than 1.

Solve: A recipe needs 2/3 cup of sugar and 1/4 cup of flour. How much sugar and flour in total?

Find LCD: 12. 2/3 = 8/12, 1/4 = 3/12. Add: 8/12 + 3/12 = 11/12 cup.

Solve: Tom walked 5/6 mile and then 1/3 mile. How far did he walk in total?

LCD is 6. 1/3 = 2/6. Add: 5/6 + 2/6 = 7/6 = 1 1/6 miles.

What is the LCD of 5/12 and 11/18?

The multiples of 12: 12,24,36,48...; multiples of 18: 18,36,54...; the smallest shared is 36.

Add: 5/12 + 11/18

LCD is 36. Rewrite: 5/12 = 15/36, 11/18 = 22/36. Add: 15/36 + 22/36 = 37/36 = 1 1/36.

What is the product of the denominators of 1/2 and 2/3? Is it the LCD?

Product is 2×3=6. Yes, 6 is the LCD for 1/2 and 2/3.

Why might you use the product of denominators instead of the LCD?

It is always a common denominator, so it works, but the numbers can be larger. For 5/12 and 11/18, the product is 216, which is bigger than 36, making the fractions harder to work with.

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