Multi-Step Problems

Multi-Step Problems

A multi-step problem chains together skills you already have: a percent change followed by another percent change, a recipe scaled up and then converted to different units, an average recomputed after a new value is added. No single step is difficult. The actual skill is completing every step in the chain instead of stopping after the first one, and the wrong answer choices are often exactly the intermediate results.

Plan before you calculate

Read the whole question and identify what is being asked for at the very end. Then work backward to list the steps. For example, "the final price after a markup and a discount" needs three numbers: the cost, the price after the markup, and the price after the discount. Writing a short line for each step keeps you from reporting an intermediate value.

Sequential percent changes

Two sequential percent changes do not combine into one simple percentage. A $20\%$ increase followed by a $10\%$ decrease is not a net $10\%$ increase. Each percent applies to the value produced by the previous step, so apply the multipliers one after the other: $1.20$, then $0.90$.

Worked example. A $200 item is increased $15\%$, then decreased $20\%$ from the new price. Find the final price.

After the increase: $200\times 1.15=230$. After the decrease: $230\times 0.80=184$. The final price is $184.

Notice that $184 is less than the starting $200 even though the increase was smaller than the decrease, which is a reminder that the $20\%$ was taken from a larger base than the $15\%$ was added to. In the same way, a markup followed by a discount, such as a store raising a cost of $100 by $35\%$ to $135 and then taking $15\%$ off, gives $135\times 0.85=114.75$, not the original price and not a $20\%$ net change. A discount followed by sales tax works the same way: multiply by $0.75$ for a $25\%$ discount, then by $1.08$ for $8\%$ tax.

Scaling and converting in the same chain

When a question asks for a value in different units from the ones given, the conversion is a required last step, not an optional afterthought. A chain that scales a recipe and then asks for a different unit, or computes a distance in miles and asks for kilometers, is unfinished until that final conversion is applied.

Worked example. A recipe for 6 servings needs 3 cups of sugar. Scaled to 15 servings, find the amount of sugar needed in tablespoons. (1 cup = 16 tablespoons)

Scale first: $3\times\frac{15}{6}=7.5$ cups. Then convert: $7.5\times 16=120$ tablespoons. The choice $7.5$ is the trap for anyone who stops after scaling.

Rates inside chains

Rate problems often carry an extra step. A pace of $8$ minutes per mile for $3.5$ miles takes $8\times 3.5=28$ minutes. A car traveling at $60$ miles per hour for $2.5$ hours covers $150$ miles, and the question may ask for the distance in kilometers: $150\times 1.60934\approx 241.4$. Read the units in the question carefully, and keep each unit attached to its number.

Averages in chains

Average questions chain through totals. Five exams averaging $85$ have a total of $425$. Adding a sixth score of $97$ makes the total $522$, so the new average is $\frac{522}{6}=87$. Reporting the old average, $85$, is the "stopped too early" mistake. A weighted score is a chain too: for scores of $75$, $88$, and $84$ weighted $20\%$, $20\%$, and $60\%$, compute each weighted piece and add: $15+17.6+50.4=83$.

Working backward through a chain

If the chain gives you the end result and asks for the start, reverse each step in the opposite order. A store marks up its cost by $40\%$ to reach a retail price of $84. Then cost $\times 1.40=84$, so cost $=\frac{84}{1.40}=60$. Reversing a percent change means dividing by its multiplier, not applying the opposite percentage.

A full example from start to finish

Suppose a store buys a shirt for $40, marks it up $50\%$, and then sells it at a $25\%$ discount from the marked price. The marked price is $40\times 1.50=60$, and the sale price is $60\times 0.75=45$, so the store's profit is $45-40=5$ dollars. Three separate numbers were needed, and the answer choices are likely to include $60 (the marked price), $45 (the sale price), and $20 (the markup). Naming the quantity you are solving for before you start prevents picking one of these intermediate results.

Strategy

• Write the steps in order before computing; circle the final quantity and unit requested.
• Apply percent changes one at a time, each to the latest value, using multipliers such as $1.15$ and $0.80$.
• Do all conversions required by the question, even when the number looks "finished."
• Use the answer choices as a check: if one of them equals your intermediate value, you probably stopped too early.
• Round only at the very end, and only as the question directs.