Integrating Essential Skills
Multi-step word problems: rates, percents, proportions and averages.
What it tests
These are multi-step problems that combine everyday math: percents, rates, ratios and proportions, averages, unit rates, and basic area and volume. The math itself isn’t advanced, but you have to set it up correctly and not stop one step early.
How it shows up
- The Math test is 45 questions in 50 minutes — a little over a minute each. Every question has four answer choices, a calculator is allowed on the whole section, there is no penalty for guessing, and no formula sheet is given.
- Integrating Essential Skills makes up a large share of the Math test — roughly 40% of the questions.
- Typical questions are word problems: discounts and tax, “how long will it take…”, map scales, mixtures of ratios, and comparing two plans or prices.
Key rules & concepts
The content you actually need to know.
Percents
- Part = percent × whole. 35% of 480 = 0.35 × 480 = 168.
- Percent change = (new − old) / old × 100%.
- Increase by r% → multiply by (1 + r). Decrease by r% → multiply by (1 − r). Write r as a decimal: 15% more → × 1.15.
- Successive changes multiply: +20% then −20% is 1.2 × 0.8 = 0.96, a 4% decrease overall.
- A percent OF a percent multiplies: 25% of 35% of the students = 0.25 × 0.35 × total.
Ratios and proportions
- Set up equal fractions with matching units and cross-multiply: a/b = c/d → ad = bc.
- Part-to-part vs. part-to-whole: a ratio of 2 : 3 means 2 parts out of 5 total, so the first part is 2/5 of the whole.
- Scale drawings and maps are proportions: (map length)/(real length) stays constant.
Rates
- Distance = rate × time. Rearrange: time = distance / rate.
- Average speed = TOTAL distance / TOTAL time — never the average of the speeds.
- Work together: add rates. If A does a job in a hours and B in b hours, together they do 1/a + 1/b of the job per hour.
- Comparing plans: set cost expressions equal to find the break-even point.
Averages and unit rates
- Weighted average: (sum of value × weight) / (total weight).
- Unit rate = price or quantity per 1 unit — the easy way to compare deals.
Strategy & traps
Step by step
- 1Write down what you’re solving for, with its units.
- 2Break the problem into mini-steps and label each number you find.
- 3Estimate first — it lets you cross out answers that are way too big or small.
- 4At the end, check units and ask: “Is this reasonable?”
Stopping one step early
The answer to your first step (like the discount amount) is often a wrong choice.
Averaging speeds
Going slower takes more time, so the average speed is closer to the slower speed.
Wrong base for a percent
A percent change is always relative to the ORIGINAL value.
Worked examples
Try each one first, then reveal the solution.
Example 1
A jacket costs $80. It is on sale for 25% off, and then an 8% sales tax is applied to the sale price. What is the final cost?
- A$60.00
- B$64.80
- C$66.40
- D$86.40
Example 2
Pump A can fill a pool in 6 hours. Pump B can fill the same pool in 3 hours. Working together, how many hours will the two pumps take to fill the pool?
- A1.5
- B2
- C4.5
- D9
Example 3
A paint color is made by mixing blue and yellow paint in a ratio of 2 : 3. How many liters of blue paint are needed to make 20 liters of the color?
- A4
- B8
- C12
- D13⅓
Practice
Pick an answer for instant feedback. Answers lock once chosen.
Question 1
Dev drives 120 miles to a lake at 60 miles per hour and returns along the same route at 40 miles per hour. What is his average speed for the whole round trip, in miles per hour?
Question 2
A store raises the price of a game by 20%. A month later, it lowers the new price by 20%. Compared with the original price, the final price is:
Question 3
On a map, 1.5 centimeters represents 12 kilometers. Two towns are 5.5 centimeters apart on the map. How many kilometers apart are the towns?
Question 4
A school has 480 students. 35% of the students are in the band, and 25% of the band members also play in the orchestra. How many students are in both the band and the orchestra?
Question 5
Phone Plan X costs $30 per month plus $0.10 per text message. Plan Y costs $45 per month with unlimited texts. For how many texts in a month do the two plans cost the same?