Math / Algebra

QuestionWhich expressions are equivalent to $\frac{1}{2}+3 c+\frac{4}{5} ?$ Select three options. $3 c+\frac{4}{5}+\frac{1}{2}$ $1+2+3 c+4+5$ $\frac{4}{5}+\frac{1}{2}+3 c$ $2+3 c+4$ $\frac{1}{2}+\frac{4}{5}+3 c$

Studdy Solution

STEP 1

What is this asking? We're looking for expressions that are the same as $\frac{1}{2} + 3c + \frac{4}{5}$ , just written in a different order or with the fractions added. Watch out! Don't get tricked by similar-looking expressions!
We're only changing the order of the terms, not the terms themselves.
Also, make sure you add those fractions correctly!

STEP 2

1. Rearrange the original expression
2. Add the fractions in the original expression
3. Check the options

STEP 3

Remember that addition is commutative, meaning we can swap the order of things we're adding without changing the result.
Think of it like rearranging furniture in a room – it's still the same furniture, just in a different spot!

STEP 4

So, $\frac{1}{2} + 3c + \frac{4}{5}$ is the same as $3c + \frac{4}{5} + \frac{1}{2}$ , or even $\frac{4}{5} + \frac{1}{2} + 3c$ !
We just shuffled the terms around.

STEP 5

Let's add the fractions in the original expression: $\frac{1}{2} + \frac{4}{5}$ .
To do this, we need a common denominator.
The smallest common denominator for 2 and 5 is 10.

STEP 6

To get a denominator of 10, we multiply the first fraction by $\frac{5}{5}$ (which is just 1, so it doesn't change the value) and the second fraction by $\frac{2}{2}$ (also 1!).

STEP 7

So, we have $\frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2} = \frac{1 \cdot 5}{2 \cdot 5} + \frac{4 \cdot 2}{5 \cdot 2} = \frac{5}{10} + \frac{8}{10}$

STEP 8

Now that we have a common denominator, we can add the numerators: $\frac{5}{10} + \frac{8}{10} = \frac{5+8}{10} = \frac{13}{10}$

STEP 9

So, our original expression, $\frac{1}{2} + 3c + \frac{4}{5}$ , is equivalent to $\frac{13}{10} + 3c$ or $3c + \frac{13}{10}$ .

STEP 10

Let's look at the options and see which ones match what we found.
We're looking for $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $3c + \frac{13}{10}$ or $\frac{13}{10} + 3c$ .

STEP 11

Option 1: $3c + \frac{4}{5} + \frac{1}{2}$ – This is a direct match with our rearranged expression!

STEP 12

Option 2: $1+2+3c+4+5$ – Nope, this has whole numbers added to $3c$ , not fractions.

STEP 13

Option 3: $\frac{4}{5} + \frac{1}{2} + 3c$ – Another match!
This is also a rearrangement of our original expression.

STEP 14

Option 4: $2 + 3c + 4$ – Again, this has whole numbers, not fractions.

STEP 15

Option 5: $\frac{1}{2} + \frac{4}{5} + 3c$ – This is the same as our original expression, just with the $3c$ moved to the end.
It's a match!

STEP 16

The equivalent expressions are: $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $\frac{1}{2} + \frac{4}{5} + 3c$ .

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QuestionWhich expressions are equivalent to $\frac{1}{2}+3 c+\frac{4}{5} ?$ Select three options. $3 c+\frac{4}{5}+\frac{1}{2}$ $1+2+3 c+4+5$ $\frac{4}{5}+\frac{1}{2}+3 c$ $2+3 c+4$ $\frac{1}{2}+\frac{4}{5}+3 c$

Studdy Solution

STEP 1

STEP 2

1. Rearrange the original expression
2. Add the fractions in the original expression
3. Check the options

STEP 3

Remember that addition is commutative, meaning we can swap the order of things we're adding without changing the result.
Think of it like rearranging furniture in a room – it's still the same furniture, just in a different spot!

STEP 4

So, $\frac{1}{2} + 3c + \frac{4}{5}$ is the same as $3c + \frac{4}{5} + \frac{1}{2}$ , or even $\frac{4}{5} + \frac{1}{2} + 3c$ !
We just shuffled the terms around.

STEP 5

Let's add the fractions in the original expression: $\frac{1}{2} + \frac{4}{5}$ .
To do this, we need a common denominator.
The smallest common denominator for 2 and 5 is 10.

STEP 6

To get a denominator of 10, we multiply the first fraction by $\frac{5}{5}$ (which is just 1, so it doesn't change the value) and the second fraction by $\frac{2}{2}$ (also 1!).

STEP 7

So, we have $\frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2} = \frac{1 \cdot 5}{2 \cdot 5} + \frac{4 \cdot 2}{5 \cdot 2} = \frac{5}{10} + \frac{8}{10}$

STEP 8

Now that we have a common denominator, we can add the numerators: $\frac{5}{10} + \frac{8}{10} = \frac{5+8}{10} = \frac{13}{10}$

STEP 9

So, our original expression, $\frac{1}{2} + 3c + \frac{4}{5}$ , is equivalent to $\frac{13}{10} + 3c$ or $3c + \frac{13}{10}$ .

STEP 10

Let's look at the options and see which ones match what we found.
We're looking for $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $3c + \frac{13}{10}$ or $\frac{13}{10} + 3c$ .

STEP 11

Option 1: $3c + \frac{4}{5} + \frac{1}{2}$ – This is a direct match with our rearranged expression!

STEP 12

Option 2: $1+2+3c+4+5$ – Nope, this has whole numbers added to $3c$ , not fractions.

STEP 13

Option 3: $\frac{4}{5} + \frac{1}{2} + 3c$ – Another match!
This is also a rearrangement of our original expression.

STEP 14

Option 4: $2 + 3c + 4$ – Again, this has whole numbers, not fractions.

STEP 15

Option 5: $\frac{1}{2} + \frac{4}{5} + 3c$ – This is the same as our original expression, just with the $3c$ moved to the end.
It's a match!

STEP 16

The equivalent expressions are: $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $\frac{1}{2} + \frac{4}{5} + 3c$ .

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Studdy Solution

STEP 1

STEP 2

1. Rearrange the original expression2. Add the fractions in the original expression3. Check the options

STEP 3

Remember that addition is *commutative*, meaning we can swap the order of things we're adding without changing the result.Think of it like rearranging furniture in a room – it's still the same furniture, just in a different spot!

STEP 4

So, 12+3c+45\frac{1}{2} + 3c + \frac{4}{5}21​+3c+54​ is the same as 3c+45+123c + \frac{4}{5} + \frac{1}{2}3c+54​+21​, or even 45+12+3c\frac{4}{5} + \frac{1}{2} + 3c54​+21​+3c!We just shuffled the terms around.

STEP 5

Let's add the fractions in the original expression: 12+45\frac{1}{2} + \frac{4}{5}21​+54​.To do this, we need a common denominator.The smallest common denominator for **2** and **5** is **10**.

STEP 6

To get a denominator of **10**, we multiply the first fraction by 55\frac{5}{5}55​ (which is just **1**, so it doesn't change the value) and the second fraction by 22\frac{2}{2}22​ (also **1**!).

STEP 7

So, we have 12⋅55+45⋅22=1⋅52⋅5+4⋅25⋅2=510+810 \frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2} = \frac{1 \cdot 5}{2 \cdot 5} + \frac{4 \cdot 2}{5 \cdot 2} = \frac{5}{10} + \frac{8}{10} 21​⋅55​+54​⋅22​=2⋅51⋅5​+5⋅24⋅2​=105​+108​

STEP 8

Now that we have a common denominator, we can add the numerators: 510+810=5+810=1310 \frac{5}{10} + \frac{8}{10} = \frac{5+8}{10} = \frac{13}{10} 105​+108​=105+8​=1013​

STEP 9

So, our original expression, 12+3c+45\frac{1}{2} + 3c + \frac{4}{5}21​+3c+54​, is equivalent to 1310+3c\frac{13}{10} + 3c1013​+3c or 3c+13103c + \frac{13}{10}3c+1013​.

STEP 10

Let's look at the options and see which ones match what we found.We're looking for 3c+45+123c + \frac{4}{5} + \frac{1}{2}3c+54​+21​, 45+12+3c\frac{4}{5} + \frac{1}{2} + 3c54​+21​+3c, and 3c+13103c + \frac{13}{10}3c+1013​ or 1310+3c\frac{13}{10} + 3c1013​+3c.

STEP 11

Option 1: 3c+45+123c + \frac{4}{5} + \frac{1}{2}3c+54​+21​ – This is a direct match with our rearranged expression!

STEP 12

Option 2: 1+2+3c+4+51+2+3c+4+51+2+3c+4+5 – Nope, this has whole numbers added to 3c3c3c, not fractions.

STEP 13

Option 3: 45+12+3c\frac{4}{5} + \frac{1}{2} + 3c54​+21​+3c – Another match!This is also a rearrangement of our original expression.

STEP 14

Option 4: 2+3c+42 + 3c + 42+3c+4 – Again, this has whole numbers, not fractions.

STEP 15

Option 5: 12+45+3c\frac{1}{2} + \frac{4}{5} + 3c21​+54​+3c – This is the same as our original expression, just with the 3c3c3c moved to the end.It's a match!

STEP 16

The equivalent expressions are: 3c+45+123c + \frac{4}{5} + \frac{1}{2}3c+54​+21​, 45+12+3c\frac{4}{5} + \frac{1}{2} + 3c54​+21​+3c, and 12+45+3c\frac{1}{2} + \frac{4}{5} + 3c21​+54​+3c.

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1. Rearrange the original expression
2. Add the fractions in the original expression
3. Check the options

Remember that addition is commutative, meaning we can swap the order of things we're adding without changing the result.
Think of it like rearranging furniture in a room – it's still the same furniture, just in a different spot!

So, $\frac{1}{2} + 3c + \frac{4}{5}$ is the same as $3c + \frac{4}{5} + \frac{1}{2}$ , or even $\frac{4}{5} + \frac{1}{2} + 3c$ !
We just shuffled the terms around.

Let's add the fractions in the original expression: $\frac{1}{2} + \frac{4}{5}$ .
To do this, we need a common denominator.
The smallest common denominator for 2 and 5 is 10.

To get a denominator of 10, we multiply the first fraction by $\frac{5}{5}$ (which is just 1, so it doesn't change the value) and the second fraction by $\frac{2}{2}$ (also 1!).

So, we have $\frac{1}{2} \cdot \frac{5}{5} + \frac{4}{5} \cdot \frac{2}{2} = \frac{1 \cdot 5}{2 \cdot 5} + \frac{4 \cdot 2}{5 \cdot 2} = \frac{5}{10} + \frac{8}{10}$

Now that we have a common denominator, we can add the numerators: $\frac{5}{10} + \frac{8}{10} = \frac{5+8}{10} = \frac{13}{10}$

So, our original expression, $\frac{1}{2} + 3c + \frac{4}{5}$ , is equivalent to $\frac{13}{10} + 3c$ or $3c + \frac{13}{10}$ .

Let's look at the options and see which ones match what we found.
We're looking for $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $3c + \frac{13}{10}$ or $\frac{13}{10} + 3c$ .

Option 1: $3c + \frac{4}{5} + \frac{1}{2}$ – This is a direct match with our rearranged expression!

Option 2: $1+2+3c+4+5$ – Nope, this has whole numbers added to $3c$ , not fractions.

Option 3: $\frac{4}{5} + \frac{1}{2} + 3c$ – Another match!
This is also a rearrangement of our original expression.

Option 4: $2 + 3c + 4$ – Again, this has whole numbers, not fractions.

Option 5: $\frac{1}{2} + \frac{4}{5} + 3c$ – This is the same as our original expression, just with the $3c$ moved to the end.
It's a match!

The equivalent expressions are: $3c + \frac{4}{5} + \frac{1}{2}$ , $\frac{4}{5} + \frac{1}{2} + 3c$ , and $\frac{1}{2} + \frac{4}{5} + 3c$ .