Math  /  Algebra

Question4 Together, Matthew and Daniel have 67 DVDs. Matthew's collection has 3 times as many DVDs as Daniel's collection. Which equation can be used to determine the number of DVDs in Daniel's collection?
A 3xx=673 x-x=67 (B) x+x=67x+x=67 (C) 3x+x=673 x+x=67 (D) 3x=673 x=67

Studdy Solution

STEP 1

What is this asking? How many DVDs does Daniel have if he and Matthew have 67 DVDs total, and Matthew has three times as many as Daniel? Watch out! Don't mix up Matthew's and Daniel's DVDs!

STEP 2

1. Set up the equation
2. Simplify and Solve

STEP 3

Let's use xx to represent the number of DVDs Daniel has.
Since Matthew has **three times** as many DVDs as Daniel, Matthew has 3x3 \cdot x DVDs.
Together, they have a **whopping** 67 DVDs.

STEP 4

So, the **total number** of DVDs is the **sum** of Daniel's DVDs (xx) and Matthew's DVDs (3x3 \cdot x).
This gives us the equation: x+3x=67x + 3 \cdot x = 67.

STEP 5

We can **simplify** our equation by combining the xx terms.
Remember, xx is really just 1x1 \cdot x, so we're adding 1x1 \cdot x and 3x3 \cdot x to get 4x4 \cdot x.

STEP 6

Now our equation looks like this: 4x=674 \cdot x = 67.
To find the value of xx, we need to **isolate** it.
Since xx is being multiplied by **4**, we'll **divide both sides** of the equation by **4**.

STEP 7

Dividing both sides by 4 gives us: 4x4=674\frac{4 \cdot x}{4} = \frac{67}{4}.
On the left side, the 4s divide to one, leaving us with just xx.
On the right side, we have 674\frac{67}{4}.

STEP 8

So, x=674x = \frac{67}{4}.
This tells us how many DVDs Daniel has.
We're not asked to calculate the fraction, but we do need to find the equation that matches our work.

STEP 9

Looking back at our simplified equation 4x=674 \cdot x = 67 from the simplification step, and comparing it to the options, we see that none of the options match exactly.
However, if we go back one step to 3x+x=673 \cdot x + x = 67, we see that this matches option (C) perfectly!

STEP 10

The equation 3x+x=673x + x = 67 or, equivalently, CC, can be used to determine the number of DVDs in Daniel's collection.

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