Math  /  Algebra

Question11 \leftarrow Suppose a is to b as c is to d ; that is, ab=cd\frac{a}{b}=\frac{c}{d}. Complete parts a through dd below. (a) Beginning with ab=cd\frac{a}{b}=\frac{c}{d}, show that bb is to aa as dd is to cc. What is the first step? A. Cross multiply. B. Add ba\frac{b}{a} to the left side and dc\frac{d}{c} to the right side. C. Subtract bb from the left side and dd from the right side. D. Subtract ba\frac{b}{a} from the left side and dc\frac{d}{c} from the right side.

Studdy Solution

STEP 1

What is this asking? If we know that ab=cd\frac{a}{b} = \frac{c}{d}, can we prove that ba=dc\frac{b}{a} = \frac{d}{c}?
And what's the very first step we should take? Watch out! Don't rush into calculations before understanding *why* those calculations make sense!
We want to manipulate our equation in a way that gets us closer to our goal!

STEP 2

1. Flip It!
2. Choose the Right First Step

STEP 3

We're given that ab=cd\frac{a}{b} = \frac{c}{d}.
Our goal is to show that ba=dc\frac{b}{a} = \frac{d}{c}.
Notice how the *numerators* and *denominators* have switched places!

STEP 4

So, how do we make that switch happen mathematically?
We can take the *reciprocal* of both sides of the equation!
This is like flipping both fractions upside down.

STEP 5

Let's do it!
Taking the reciprocal of both sides gives us: 1ab=1cd \frac{1}{\frac{a}{b}} = \frac{1}{\frac{c}{d}} which simplifies to: ba=dc \frac{b}{a} = \frac{d}{c} Boom! We've shown what we needed to!

STEP 6

Now, let's look at the multiple-choice options for the *first* step.
We just saw that the correct first step is to take the reciprocal of both sides of the original equation.
Which option matches that?

STEP 7

Option A suggests *cross-multiplying*.
That would give us ad=bca \cdot d = b \cdot c.
This isn't wrong, but it doesn't directly lead us to ba=dc\frac{b}{a} = \frac{d}{c}.

STEP 8

Options B, C, and D involve adding or subtracting terms.
These operations don't help us flip the fractions, so they're not the right first steps either.

STEP 9

Since none of the options explicitly say "take the reciprocal", we need to think about what taking the reciprocal actually *does*.
Remember, taking the reciprocal of a fraction is the same as dividing **one** by that fraction.
For example, the reciprocal of ab\frac{a}{b} is 1ab=ba\frac{1}{\frac{a}{b}} = \frac{b}{a}.

STEP 10

Since none of the options match taking the reciprocal directly, and we've shown that taking the reciprocal is the correct first step, none of the provided options are the correct first step.

STEP 11

We've successfully shown that if ab=cd\frac{a}{b} = \frac{c}{d}, then ba=dc\frac{b}{a} = \frac{d}{c}.
None of the given options describe the correct first step, which is to take the reciprocal of both sides of the equation.

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