Lesson 12: Solving Problems about Percent Increase or Decrease

Solving Problems about Percent Increase or Decrease

Timing the Relay Race

5 min

Problem

The track team is trying to reduce their time for a relay race. First, they reduce their time by 2.1 minutes. Then they are able to reduce that time by $\frac{1}{10}$ . If their final time is 3.96 minutes, what was their beginning time? Show or explain your reasoning.

Answer

6.5 minutes. Sample reasoning:

With equation: $0.9(x-2.1) = 3.96$ , $x-2.1=4.4$ , $x=6.5$ .
Reasoning with or without a diagram: 9 out of 10 parts represent 3.96 minutes, so the $\frac{1}{10}$ reduction was $3.96\div9$ or 0.44 minutes. That makes the time before the 2.1 minute reduction $3.96+0.44$ or 4.4 minutes. The original time was $4.4+2.1$ , or 6.5 minutes.

Sample Response

6.5 minutes. Sample reasoning:

With equation: $0.9(x-2.1) = 3.96$ , $x-2.1=4.4$ , $x=6.5$ .
Reasoning with or without a diagram: 9 out of 10 parts represent 3.96 minutes, so the $\frac{1}{10}$ reduction was $3.96\div9$ or 0.44 minutes. That makes the time before the 2.1 minute reduction $3.96+0.44$ or 4.4 minutes. The original time was $4.4+2.1$ , or 6.5 minutes.

Responding to Student Thinking

Press Pausepress_pause

Response: By this point in the unit, students should have some mastery of interpreting problems and solving equations involving variables and fractional amounts. If most students struggle, make time to revisit related work in the lessons referred to here. See the Course Guide for ideas to help students re-engage with earlier work.

Standards

Addressing

7.EE.2·Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."
7.EE.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
7.EE.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.A.2·Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, $a + 0.05a = 1.05a$ means that “increase by $5\%$” is the same as “multiply by $1.05$.”
7.EE.B.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
7.EE.B.4.a·Solve word problems leading to equations of the form $px + q = r$ and $p(x + q) = r$, where $p$, $q$, and $r$ are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is $54$ cm. Its length is $6$ cm. What is its width?