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Solve multi-step problems involving integers

Multi-step integer problems require interpreting positive and negative quantities, selecting operations that represent successive changes or relationships, and evaluating expressions with parentheses and the conventional order of operations. Reasoning includes applying sign rules for addition, subtraction, multiplication, and division, maintaining the meaning of intermediate results, and checking whether the final value is reasonable in context; the scope is integer arithmetic rather than general rational, real-number, or symbolic algebraic expressions.

Detailed Explanation: Solve multi-step problems involving integers

When solving a multi-step integer problem:

  1. Translate each situation into an integer. Increases are positive; decreases are negative.
  2. Write an expression that shows the steps.
  3. Use the order of operations: multiplication or division before addition or subtraction.
  4. Check whether the answer makes sense in the situation.

Example:
The temperature is −7∘C-7^\circ\text{C}. It rises 12∘C12^\circ\text{C}, then falls 3∘C3^\circ\text{C} each hour for 44 hours. What is the final temperature?

A rise of 1212 is +12+12. A fall of 33 each hour for 44 hours is 4(−3)4(-3).

Write the expression:

−7+12+4(−3)-7+12+4(-3)

First, multiply:

4(−3)=−124(-3)=-12

Now evaluate from left to right:

−7+12+(−12)-7+12+(-12)

Combine −7-7 and 1212:

−7+12=5-7+12=5

Then subtract 1212:

5+(−12)=−75+(-12)=-7

So, the final temperature is

−7∘C.\boxed{-7^\circ\text{C}}.

This is reasonable because the temperature first rises to 5∘C5^\circ\text{C}, then drops 12∘C12^\circ\text{C}, ending at −7∘C-7^\circ\text{C}.

Learn by doing: Solve multi-step problems involving integers

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