Question

Which of the following are true statements?
If a * b = c, and a is not equal to zero, then b = c / a
If a * b = c, and b is not equal to zero, then a = c / b
If m / n = p, then m = p * n
All of these statements are true.

Answer 1

Answer:

  (d)  All of these statements are true.

Step-by-step explanation:

You want to know which statements relating multiplication and division are true.

Multiplicative inverse

The multiplicative inverse of (non-zero) 'a' is 1/a. The product of these is 1 (by definition).

Multiplicative identity

1 is the multiplicative identity element, so 1·a = a.

Multiplication property of equality

Starting with a·b = c, we can multiply both sides of the equation by 1/a without altering its truthfulness.

  (1/a)(a)(b) = (1/a)(c)

  (a/a)(b) = c/a . . . . . . . for a ≠ 0

  b = c/a

Using the other factor, we have ...

  a = c/b . . . . . . . from (1/b)(a)(b) = (1/b)(c)

Substitution property

We can always substitute equals for each other. Then for c=m, a=n, b=p, the first of these equations is ...

  p = m/n

  m/n = p . . . . symmetric property of equality

Effectively, all of the multiplication and division relations shown are true:

• ab = c   ⇔   b = c/a . . . . . . . a ≠ 0
• ab = c   ⇔   a = c/b . . . . . . . b ≠ 0
• m/n = p   ⇔   m = pn  . . . . . n ≠ 0

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Additional comment

In the first two cases, we included the caveat that the divisor could not be zero. This is also true of the last case (m/n=p). However, in this case, we assume that p is defined, which automatically means that n ≠ 0.

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