Comparing Multi Step Options

Welcome back to Multi-Step Percent Problems! This is the fifth and final lesson of the course, which means you have worked through every major building block: choosing the correct base, recovering an original amount, applying consecutive percent changes, and mixing percent adjustments with fixed-dollar amounts. That is an impressive run, and you are about to tie it all together.

In everyday life, we rarely face just one offer in isolation. Instead, we are asked to choose: Which deal saves the most money? Which plan costs less over a year? Which payment option gets us ahead faster? Each choice typically involves its own chain of percent and dollar steps, and the winner is not always the one that "looks" better at first glance. In this lesson, we will learn a reliable method for calculating the final result of two competing multi-step options and selecting the one that best meets a stated goal.

Before we jump into numbers, let's frame the strategy. As you have seen throughout this course, every multi-step percent problem boils down to performing each adjustment in the correct order on the correct base. Comparing two options simply means doing that process twice — once for each option — and then placing the two final results side by side.

The critical habit is to finish all the math before you judge. A 30% discount sounds bigger than 20% off plus a $15 coupon, but whether it actually saves more money depends on the starting price and the order of steps. Our job is to let the numbers decide, not our first impression. This "calculate, then compare" mindset will guide every example in this lesson.

Let's start with a straightforward shopping example.

Scenario. A pair of headphones is priced at $120. Two stores sell the same item:

• Store A: 25% off, then a $10 coupon applied to the reduced price.
• Store B: $20 coupon first, then 15% off the remaining balance.

Our goal is the lowest total cost.

For Store A, the percent step comes first. The discount is 120 × 0.25 = $30, bringing the price to 120 − 30 = $90. Then the $10 coupon is subtracted: 90 − 10 = $80.

For Store B, the fixed-dollar step comes first. Subtracting the $20 coupon gives 120 − 20 = $100. Then the 15% discount is 100 × 0.15 = $15 off, so the final price is 100 − 15 = $85.

Store A wins by $5. Notice that Store B's 15% discount is applied to a smaller base ($100 instead of $120), which limits its impact. This is exactly why running both calculations is essential — the larger-sounding percent does not always produce the bigger savings.