Explain how to justify the Cross Products Property by using the Multiplication Property of Equality.
The Multiplication Property of Equality states that if you multiply both sides of an equation by the same number, the sides remain equal (i.e. equality is preserved).
and
Cross Product Property states that in a proportion, product of the means is equal to the product of the extremes.
So, per the Cross Product property:
A/B = C/D
So we're trying to prove that AD=BC
You multiply both sides by B. You get AB over B which is A and CD over D. So now you have A=CB/D
Now multiply both sides by D. You get AD = CBD over D which is the same as AD = BC.
Ta-dah!
To explain how to justify the Cross Products Property using the Multiplication Property of Equality, let's first define what each property means:
1. Cross Products Property: This property states that for any proportion of the form a/b = c/d, the product of the means (b * c) is equal to the product of the extremes (a * d). Mathematically, it can be written as bd = ac.
2. Multiplication Property of Equality: This property states that for any equation a = b, if you multiply both sides of the equation by the same non-zero number, the equality still holds. Mathematically, it can be written as a * c = b * c.
Now, let's see how we can use the Multiplication Property of Equality to justify the Cross Products Property.
Consider a proportion a/b = c/d, where a, b, c, and d are non-zero values.
To justify the Cross Products Property, we can start by multiplying both sides of the proportion by bd (the product of the denominators):
(a * bd) / b = (c * bd) / d
Now, we can simplify the expression:
ab * d/ b = cd * b/ d
Notice that the b/d and d/b ratios are equal to 1 because dividing a number by itself always yields 1. Therefore, we can rewrite the expression as:
ab * 1 = cd * 1
Which simplifies to:
ab = cd
Now, we have successfully arrived at the equation ab = cd, which is the Cross Products Property. This justifies the usage of the Cross Products Property based on the Multiplication Property of Equality.
By using the Multiplication Property of Equality, we multiplied both sides of the proportion by bd to eliminate the denominators and simplify the equation. This process allowed us to obtain an equation that is equivalent to the Cross Products Property.