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Let m be the number of math teachers in a school, f the number of foreign language teachers, s the number of science teachers, and a the number of art teachers. For each statement below, write three equations that demonstrate a true relationship between the variables.


AThere are twice as many math teachers in a school as foreign language teachers.
BThere are 3 fewer science teachers than math teachers.
CThere are 5 more foreign language teachers than half the number of art teachers.

Verified Answer :

  • For statement (a): \(m = 2f\), \(f = \frac{m}{2}\), \(m - 2f = 0\).
  • For statement (b): \(s = m - 3\), \(m = s + 3\), \(m - s = 3\).
  • For statement (c): \(f = \frac{a}{2} + 5\), \(f - 5 = \frac{a}{2}\), \(a = 2f - 10\).
  • These equations represent the same relationships between variables using algebraic rearrangement. \(\boxed{\text{See step-by-step logic.}}\)

Explanation

  1. Data and Problem Analysis

The problem requires defining variables for different categories of teachers: \(m\) for math, \(f\) for foreign language, \(s\) for science, and \(a\) for art. We are given three specific statements defining relationships between these variables and need to translate each statement into three different mathematical expressions.

  1. Translating Statement (a)

For the statement "There are twice as many math teachers in a school as foreign language teachers", we start with the primary relationship \(m = 2f\). By rearranging this algebraic equation, we can obtain two additional equivalent expressions.

  • Primary: \(m = 2f\)
  • Rearranged: \(f = \frac{m}{2}\)
  • Rearranged: \(m - 2f = 0\)
  1. Translating Statement (b)

For the statement "There are 3 fewer science teachers than math teachers", we start with the primary relationship \(s = m - 3\). By rearranging this algebraic equation, we can obtain two additional equivalent expressions.

  • Primary: \(s = m - 3\)
  • Rearranged: \(m = s + 3\)
  • Rearranged: \(m - s = 3\)
  1. Translating Statement (c)

For the statement "There are 5 more foreign language teachers than half the number of art teachers", we start with the primary relationship \(f = \frac{a}{2} + 5\). By rearranging this algebraic equation, we can obtain two additional equivalent expressions.

  • Primary: \(f = \frac{a}{2} + 5\)
  • Rearranged: \(f - 5 = \frac{a}{2}\)
  • Rearranged: \(a = 2(f - 5) = 2f - 10\)

Examples

In school administration, these algebraic relationships are essential for resource allocation. For example, if a principal knows that the number of art teachers is \(a = 10\), they can use the relationship \(f = \frac{a}{2} + 5\) to determine that they need \(f = 5 + 5 = 10\) foreign language teachers. By writing these relationships as equations, school leaders can quickly solve for any missing staffing information as the number of students or curriculum requirements changes.

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