Mixed Multi-Step Word Problems

This capstone lesson pulls together fractions, decimals, percents, ratios, rates, motion, work and money. You will practice reading a long word problem, deciding what it really asks, splitting it into small steps, keeping units consistent, estimating before calculating, and checking that the final number answers the question that was asked.

Multi-step problems are rarely hard because of the arithmetic. The difficulty is that one result feeds the next, so an early slip carries through, and the wording may include extra facts or ask for something other than the obvious number. Test writers also put intermediate results among the answer choices. A steady routine and careful labeling remove most of these risks.

TARGET: state exactly what the question wants, such as "dollars left" or "minutes after B starts." DATA: list the numbers that matter. UNITS: note each unit and convert to a common one. MODEL: write the operations or equation before computing. SOLVE: work step by step, labeling each result. CHECK: confirm the answer is reasonable and matches the target.

Label each result so you know whether it is a stepping stone or the answer. A depot has 600 crates. It ships 40% on Monday (240), then half of what is left on Tuesday (180). The 360 crates left after Monday is only an intermediate value; the final answer is 180 crates remaining.

Word problems often include facts you do not need, such as the driver's name, the color of the truck, or the day of the week. Before computing, match each number to the target. If a question asks for fuel cost, you need distance, mileage and price per gallon; the number of passengers is a distraction.

You can only add or compare quantities in the same unit. A cable of 2 yards plus a 4-foot piece is

not 6 of anything. A task lasting 1.5 hours followed by one of 40 minutes takes

Choose one unit, convert everything to it, then combine.

The story decides the order of operations. If each of 5 boxes costs 8 dollars plus 2 dollars shipping, the total is

Writing

charges shipping only once. Ask whether each charge applies per item or once per order.

Split a long problem into blocks that each answer one small question, such as "How many were made?", "How many were good?", "What did they cost?" Finish one block, write its labeled result, and feed it into the next. Short blocks are easy to check.

Apply each fraction or percent to the correct group. Of 720 recruits, $\frac{5}{8}$ pass a first test:

If 80% of those 450 pass a second test, 360 recruits pass both. The 80% applies to 450, not to 720.

For a ratio and a total, add the ratio parts to find how many equal shares the total is split into. With a ratio of 3:5 and a total of 64, there are 8 parts:

Each part is 8, so the groups are

First use the rate to get a total, then apply the percent. A line makes 150 units per hour for 8 hours, or 1,200 units. If 4% are defective, that is 48 units, leaving 1,152 good units.

Find the time for each driving leg, then add any stops. A convoy drives 180 miles at 60 mph (3 hours), takes a 45-minute break, then drives 100 miles at 50 mph (2 hours). Total time: 5.75 hours, or 5 hours 45 minutes. Leaving at 7:00 a.m., it arrives at 12:45 p.m.

A crew that needs 10 days alone works 4 days, finishing 2/5 and leaving 3/5. A second crew that needs 15 days then joins. Together they do

per day, so the rest takes

more days.

Multiply the unit price by the quantity, then add any flat charges. Fencing costs 7.25 dollars per meter, and a homeowner needs 18 meters:

Adding a one-time 45-dollar installation fee gives 175.50 dollars. A flat fee is added once, not per meter.

Some problems give the result and ask for the starting value. Undo each step in reverse order. After a 15% raise, a wage is 23 dollars per hour. The new wage is 1.15 times the old, so the old wage was

Subtracting 15% of 23 would be wrong.

Round the numbers and get a rough answer before calculating. If 6 items at about 12 dollars cost roughly 72 dollars, a choice of 7.20 or 720 dollars is clearly wrong. An estimate also tells you whether a result should be larger or smaller than a given value.

Answer choices can guide you. Eliminate any that fail a quick estimate or have the wrong unit or size. Notice which choices are intermediate values, since one is often the result after only some of the steps. In reverse problems, you can test a choice by running it forward. Do not choose until your result answers the exact target.

Problem: An armory holds 960 rounds. One-fourth go to Range A. Then $\frac{3}{8}$ of the rest go to Range B. How many rounds remain?

Step 1: Find one-fourth of 960 for Range A. To take one-fourth, divide by 4.

Step 2: Subtract to find what is left after Range A. This is "the rest" that the next step talks about.

Step 3: Find $\frac{3}{8}$ of 720 for Range B. Divide by 8 to get one-eighth, then multiply by 3.

Step 4: Subtract Range B's share from the 720 that were left.

Answer: 450 rounds remain.

Check: Add all three groups. They should make the full 960.

Problem: A plant runs two shifts of 7.5 hours each, making 120 units per hour. Three percent are defective. Good units are packed 24 per box. How many full boxes are filled?

Step 1: Find the total hours worked. There are 2 shifts of 7.5 hours.

Step 2: Find the total units made. Multiply hours by units per hour. Split 120 into 100 + 20 to make it easy.

Step 3: Find 3% of 1,800 to get the defective units. 1% means divide by 100, then multiply by 3.

Step 4: Subtract the defective units to get the good units.

Step 5: Divide the good units by 24 to count boxes. Take out 70 boxes first, then see what is left.

So

with a remainder of 18. The 18 extra units are not enough for another full box.

Answer: 72 full boxes.

Check: 72 boxes of 24 plus the 18 extras should give back 1,746.

Problem: Six filters cost 12.50 dollars each. A 20% discount applies, then 7% tax. What is the final cost?

Step 1: Find the subtotal. Multiply the price by 6. Think of 12.50 as 12 + 0.50.

The subtotal is 75 dollars.

Step 2: Take off the 20% discount. 10% of 75 is 7.50, so 20% is twice that.

After the discount, the price is 60 dollars. (This is the same as 0.80 × 75.)

Step 3: Add 7% tax on the discounted price. 1% of 60 is 0.60, so 7% is 7 times that.

This is the same as 60 × 1.07.

Answer: The final cost is 64.20 dollars.

Check: The discount took off a fifth (about 15 dollars), and the tax added back a small amount (about 4 dollars). 75 − 15 + 4 is about 64, which matches.

Problem: A driver travels 150 miles at 60 mph, stops 30 minutes, then travels 100 miles at 50 mph. What is the overall average speed, including the stop?

Step 1: Find the time for the first leg. Time = distance ÷ speed.

Step 2: Change the stop into hours. 30 minutes is half of 60 minutes.

Step 3: Find the time for the second leg.

Step 4: Add all the times. The stop counts, because the question says "including the stop."

Step 5: Add the distances.

Step 6: Average speed = total distance ÷ total time.

Answer: 50 mph.

Check: Driving 50 mph for 5 hours covers

the full trip. Do not just average 60 and 50; that ignores the stop and the different times.

Problem: A unit of 132 people has officers and enlisted members in a 2:9 ratio. Two-thirds of the enlisted are on duty. How many enlisted are on duty?

Step 1: Add the ratio parts to find how many equal parts make the whole unit.

Step 2: Find the size of one part. Divide the total people by the number of parts.

Step 3: Enlisted members are 9 parts.

Step 4: Find $\frac{2}{3}$ of 108. Divide by 3, then multiply by 2.

Answer: 72 enlisted members are on duty.

Check: Officers are

the whole unit. Also 72 is less than 108, as it should be.

Problem: A mechanic can finish a job in 6 hours and works alone for 2 hours. A coworker who can do the whole job in 3 hours then helps. How much longer will it take?

Step 1: Find the mechanic's rate. The whole job takes 6 hours, so each hour does $\frac{1}{6}$ of the job.

Step 2: Find how much is done in 2 hours, and how much remains.

Step 3: Find the team rate. The coworker does $\frac{1}{3}$ of the job per hour. Add the rates using sixths.

Step 4: Time = work left ÷ rate. Dividing by $\frac{1}{2}$ is the same as multiplying by 2.

Step 5: Change $\frac{4}{3}$ hours to hours and minutes. It is 1 hour plus $\frac{1}{3}$ hour, and $\frac{1}{3}$ of 60 minutes is 20 minutes.

Answer: $\frac{4}{3}$ hours, or 1 hour 20 minutes.

Check: Working together at $\frac{1}{2}$ job per hour for $\frac{4}{3}$ hours does

exactly the work that was left.

Problem: After a 25% discount, a pair of boots costs 54 dollars. What was the original price?

Step 1: Figure out what percent of the original is left. Taking off 25% leaves 75% of the original.

Step 2: Write an equation. 75% of the original price is 54 dollars.

Step 3: Divide 54 by 0.75. Since 0.75 = $\frac{3}{4}$, dividing by it is the same as multiplying by $\frac{4}{3}$: divide by 3, then multiply by 4.

Answer: The original price was 72 dollars.

Check: 25% of 72 is one-fourth of 72, which is 18. Then

the sale price.

Watch out: Adding 25% to 54 (giving 67.50) is wrong, because the discount was 25% of the original price, not of the sale price.

Stopping one step early: choosing the amount used instead of the amount remaining, or the discount instead of the sale price. Recheck the target.

Applying a percent to the wrong group: a percent of "those assigned" means the assigned count, not the whole.

Mixing units: adding minutes to hours or feet to yards without converting.

Reversing percents by adding or subtracting: undo a change by dividing by the multiplier.

Averaging speeds instead of using total distance over total time.

Charging a flat fee per item, or a per-item fee only once.

Before choosing, take twenty seconds. Reread the last sentence of the question and confirm your number matches what it asks. Check that the unit is right. Ask whether the size makes sense: a remainder smaller than the start, a team faster than one worker, a sale price below the original. Make sure you did not stop on an intermediate value, then select the answer.

Arithmetic Reasoning combines a few core tools. Fractions and percents scale a quantity, and each applies to a specific base. Ratios split a total into equal parts. Unit rates come from division and scale by multiplication. Motion uses

with relative speed for two travelers and total distance over total time for averages. Work rates add as fractions of one job. Money problems use discounts, tax, wages, commission and simple interest. Apply them in labeled steps, convert units, estimate, and check the target.