How to find the average speed over the whole trip?

If the distances are equal, the average speed is the harmonic mean: v_avg = 2·v₁·v₂/(v₁+v₂). You cannot simply average the speeds (60+40)/2 = 50 — this is a typical mistake.

How to solve problems on working together?

Each worker’s rate is 1/time (the share of work per hour). Add the rates to get the combined rate. The time to complete the work together = 1 / combined rate.

How to find the new price after a discount?

First find the discount amount: old price × percentage / 100. Then subtract the discount from the old price. For example, 800 − 25% of 800 = 800 − 200 = 600.

How to find the concentration of a mixture of two solutions?

Find the total amount of “pure substance” (volume × concentration for each solution), add them, then divide by the total volume of the mixture. Multiply by 100% if the concentration in percent is needed.

Word problems

This page is part of the section “USE basic level. Word problems”. Here are solutions to typical problems on motion, work, percentages and mixtures. All problems are written independently, with neat answers.

Problems with solutions

Problem 1. Average speed

A car travels from A to B at 60 km/h and back at 40 km/h. Find the average speed of the car over the whole trip.

The average speed over equal distances (there and back) is given by the harmonic mean:

v_{ср} = \frac{2 \cdot v_{1} \cdot v_{2}}{v_{1} + v_{2}} = \frac{2 \cdot 60 \cdot 40}{60 + 40} = \frac{4800}{100} = 48

Answer: 48 км/ч.

Problem 2. Working together

The first worker completes an order in 6 hours, the second in 12 hours. How many hours will they take to complete the order working together?

The first worker’s rate is 1/6 of the order per hour, the second’s is 1/12. Add the rates:

\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}

Together they complete 1/4 of the order per hour. So the whole order will take:

t = \frac{1}{1 / 4} = 4

Answer: 4 ч.

Problem 3. Percentage discount

A product cost 800 rubles. The price was reduced by 25%. How much does the product cost now?

Find 25% of 800 — this is the discount amount:

800 \cdot \frac{25}{100} = 200

800 - 200 = 600

Answer: 600 руб.

Problem 4. Concentration of a mixture

3 liters of a 20% solution and 2 liters of a 30% solution are mixed. Find the concentration of the resulting mixture.

Find the total amount of “pure substance” in both solutions:

3 \cdot 20 + 2 \cdot 30 = 60 + 60 = 120

The total volume of the mixture is 3 + 2 = 5 L. The concentration of the mixture:

\frac{120}{3 + 2} = \frac{120}{5} = 24

Answer: 24%.