Arithmetic (Beginner) · Quick Reference

Mixture Problems Cheat Sheet

Arithmetic (Beginner) · Lesson 21/21

In one line: mixture problems combine two or more quantities with different properties (like concentration or price) to find a resulting average or unknown proportion.

Key Ideas

1Total Value Method. Total value of mixture = sum of (quantity x property) of each part.

2Alligation Rule. A shortcut to find the ratio in which two ingredients must be mixed to get a desired average, based on how far each ingredient's value is from the mean.

3Concentration Problems. When mixing solutions of different concentrations, track the total amount of the 'pure' substance separately from the total volume.

4Replacement Problems. When part of a mixture is removed and replaced with another substance, track what fraction of the original substance remains after each replacement.

Worked Examples

In what ratio should tea worth $40/kg be mixed with tea worth $60/kg to get a mixture worth $50/kg?

1:1

How many liters of a 20% acid solution must be mixed with a 50% acid solution to make 30 liters of a 30% solution?

20 L of 20% solution and 10 L of 50% solution

A shopkeeper mixes 20 kg of rice at $30/kg with 10 kg of rice at $45/kg. Find the price per kg of the mixture.

$35/kg

Formulas

Alligation: Ratio of quantities = (d2 - Mean) : (Mean - d1), where d1 and d2 are the two given values

Practice Yourself

Mix rice at $20/kg and $30/kg to get a mixture at $25/kg. Find the ratio.

1:1

Find the average price mixing 5 kg at $10/kg with 5 kg at $20/kg.

$15/kg

Mix nuts at $5/kg and $8/kg to get a mixture at $6/kg. Find the ratio.

2:1