Case File: How to Solve Multi-step Problems Involving Percent

1. Case Background: Understanding Multi-step Problems Involving Percent

Before we work through the investigation, let’s brief ourselves on the terms involved:

• Multi-step Problems: These are problems that require more than one operation or step to find the solution.
• Percent: A ratio or fraction expressed out of \(100\).

2. The Investigation: How to Solve Multi-step Problems Involving Percent

Now that we’re armed with the basics, it’s time to jump into the mystery!

Detective’s Guide: Solving Multi-step Problems Involving Percent

Step 1: Understand the Problem

First, read the problem carefully. Identify what you’re asked to find and what information you have.

Step 2: Make a Plan

Develop a strategy to solve the problem. This could involve determining percentages, adding or subtracting amounts, or other operations.

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Step 3: Carry Out the Plan

Execute your strategy. Remember, you might have to perform several steps to find the solution.

Step 4: Check Your Solution

Finally, review your work to ensure your answer makes sense in the context of the problem.

For example, if you buy a book that costs \($25\) and gets a \(20\)\(\%\) discount and has to pay \(5\)\(\%\) sales tax, what’s the final price?

1. Understand the Problem: You need to find the final price after applying a discount and tax to a \($25\) book.
2. Make a Plan: First, find the discounted price. Then, apply the tax.
3. Carry Out the Plan:
   • The discount is \(20\)\(\%\) of \($25\), or \($5\). So, the discounted price is \($25\ – $5 = $20\).
   • The tax is \(5\)\(\%\) of \($20\), or \($1\). So, the final price is \($20+ $1 = $21\).
4. Check Your Solution: A \($25\) book with a \($5\) discount and \($1\) tax should cost \($21\).

Well done, detectives! You’ve solved the mystery of multi-step problems involving percent. Always remember, no case is too tough when you break it down into manageable steps. Keep your wits about you, and happy sleuthing!

Recommended EffortlessMath Books

For a step-by-step build of every percent skill from the ground up, the Pre-Algebra for Beginners covers percents, ratios, and proportions with worked examples and practice sets. If you’re prepping for a standardized test, the Algebra I for Beginners includes multi-step percent word problems with full solutions.

More Multi-Step Percent Problems, Solved Step by Step

Example 1: Two discounts in a row

A $120 jacket is 30% off, and you get an extra 10% off the sale price. Step 1: 120 × 0.70 = $84. Step 2: 84 × 0.90 = $75.60. The total discount is $44.40, which is 37% of $120, not 40%. The second discount is taken from a smaller number.

Example 2: Discount, then tax

An $80 game is 25% off, and sales tax is 6%. Step 1: 80 × 0.75 = $60. Step 2: 60 × 1.06 = $63.60. You pay $63.60.

Example 3: Base pay plus commission

A salesperson earns $400 a week plus 5% commission on sales. This week she sold $6,000. Step 1: 5% of 6,000 = 0.05 × 6,000 = $300. Step 2: 400 + 300 = $700.

Example 4: Working backward from a total with tax

A restaurant bill is $64.20 after 7% tax. What was the bill before tax? The total is 107% of the original, so divide: 64.20 ÷ 1.07 = $60. Check: 60 × 1.07 = 64.20.

Try It Yourself: Multi-Step Percent Practice

1. A $45 pair of shoes is 20% off, then 8% tax is added. What is the final price?
2. A town of 5,000 people grows by 10% one year and 10% the next. What is the new population?
3. A store marks up a $40 item by 50%, then puts it on sale for 30% off. What is the sale price?
4. After a 15% discount, a bike costs $170. What was the original price?
5. In a class of 30 students, 60% are girls. What is the ratio of girls to boys?

Answers: 1) 45 × 0.80 = 36, then 36 × 1.08 = $38.88. 2) 5,000 × 1.10 = 5,500, then 5,500 × 1.10 = 6,050 people. 3) 40 × 1.50 = 60, then 60 × 0.70 = $42. 4) 170 ÷ 0.85 = $200. 5) 18 girls and 12 boys, so 18:12 = 3:2.

Frequently Asked Questions

What is a multi-step percent problem?

A percent problem that needs two or more operations to solve. Example: “A jacket is \(\$80\). It’s 25% off, then 8% sales tax is added. What’s the final price?” You compute the discount, subtract it, then compute the tax on the discounted price – three steps strung together.

How do I find a percent of a number?

Convert the percent to a decimal (move the decimal point two places left, or divide by 100), then multiply. Example: 15% of 60 is \(0.15 \times 60 = 9\). For percents like 50%, 25%, and 10%, the mental shortcuts are halves, quarters, and tenths – faster than converting to decimals.

What’s the formula for percent change?

Percent change \(= \dfrac{\text{new} – \text{old}}{\text{old}} \times 100\). If the result is positive, it’s an increase; if negative, a decrease. Example: from 50 to 65, the change is \(\dfrac{65-50}{50} \times 100 = 30\%\) increase.

How do I handle discount-then-tax problems?

Two steps. First, find the discounted price: original price minus (percent off times original price). Second, find the tax: percent tax times the discounted price. Then add the tax to the discounted price. Don’t apply tax to the original price – that’s a common slip.

Do two 10% increases equal a 20% increase?

No. Two consecutive 10% increases give you a 21% total increase, because the second 10% is taken on the larger value. \(100 \to 110 \to 121\), not \(100 \to 120\). Percent changes don’t simply add when they’re applied to changing bases.

Do a 20% increase and a 20% decrease cancel out?

No. If you start with 100 and increase by 20%, you get 120. Then decrease 120 by 20% (which is 24), and you get 96 – not 100. The percents act on different bases, so they don’t cancel. This is a classic trap on standardized tests.

How do I work backwards from a final price?

Set up an equation. If a jacket is \(\$48\) after a 20% discount, then \(\$48\) is 80% of the original (\(100\% – 20\%\)). So \(\text{original} = 48 / 0.80 = \$60\). Always set up: final amount equals the remaining percent (as a decimal) times the original.

How does sales tax work in percent problems?

Sales tax is added to the price, so the final price is \(\text{price} \times (1 + \text{tax rate})\). For a \(\$50\) item with 7% tax, the final price is \(50 \times 1.07 = \$53.50\). The trick is making sure you apply tax to the right number – usually the price AFTER any discount.

What’s a tip-and-split problem?

Common form: a restaurant bill is \(\$60\), you add a 20% tip, then split the total among 4 people. Step 1: tip is \(0.20 \times 60 = \$12\), so total is \(\$72\). Step 2: each person pays \(72/4 = \$18\). Keep the calculation in order; the split happens AFTER the tip.

Where do multi-step percent problems show up on tests?

Everywhere – SAT, ACT, GED, HiSET, GRE, Praxis Core, ASVAB, and almost every state grade-level math test from grade 6 up. Common scenarios: discounts and tax, tipping, commission, simple interest, percent change over multiple time periods. Practice spotting which base each percent acts on – that’s the main skill being tested.

How do you solve multi-step ratio problems?

Break the problem into one relationship at a time. First find the size of one part, then use it to find the amount you need. For example, if pens and pencils are in a 3:5 ratio and there are 96 in all, there are 8 parts, so one part is 12. That gives 36 pens and 60 pencils.

Can you give me some examples of multi-step percent problems?

Common examples are a sale price followed by sales tax, two discounts in a row, a tip on a bill that is then split among friends, a population that grows by a percent each year, and pay that includes a percent commission. Each one needs at least two percent calculations, done in order.

Is 20% off then another 10% off the same as 30% off?

No. The second discount is taken from the already lowered price. On a $100 item, 20% off gives $80, and 10% off $80 gives $72. That is a total discount of 28%, not 30%. To combine discounts, multiply the decimals: 0.80 × 0.90 = 0.72.

How do I check my answer to a multi-step percent problem?

Estimate first. If something is on sale and then taxed, the final price should be a little more than the sale price but less than the original. You can also work the problem backward: divide by each multiplier in reverse order and see if you get back to the starting amount.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• How to find the percent of a number
• How to solve percent of change
• How to solve discount, tax, and tip word problems
• How to solve simple interest problems
• How to solve multi-step word problems