Overview

The main skill is translation: identify variables, write equations, and keep units consistent. Draw tables for rate and mixture problems.

Most errors come from unclear variables or inconsistent units, not algebra.

Key Ideas

• Distance = rate $\times$ time.
• For mixtures, total amount = sum of parts, and total concentration is a weighted average.
• Ratios are easiest to manipulate when written as fractions.
• Percent changes should be applied multiplicatively, not additively.
• Use tables for work and rate problems to avoid sign mistakes.

Core Templates

Rates and Work

If an agent completes a job in $t$ hours, its rate is $1/t$ jobs per hour. When multiple agents work together, add rates.

Mixtures

Amount of solute $=$ concentration $\times$ total volume. Track solute, not just volume.

Ratios

Translate $a:b$ as $a/b$ or as $a=kx$, $b=ky$ to scale cleanly.

Percent Change

Increase by $p\%$: multiply by $1+p/100$. Decrease by $p\%$: multiply by $1-p/100$.

Worked Example

A tank is filled by two pipes. Pipe A fills the tank in $6$ hours, pipe B in $10$ hours. How long together?

Their combined rate is $\frac{1}{6} + \frac{1}{10} = \frac{8}{30} = \frac{4}{15}$ tanks per hour. The time is $\frac{1}{4/15} = \frac{15}{4}$ hours.

More Examples

Example 1: Distance

Two cars are 120 miles apart and drive toward each other at 50 mph and 30 mph. How long until they meet?

Relative speed is $80$ mph, so time is $120/80 = 1.5$ hours.

Example 2: Mixture

A solution is 20% salt. If you add 10 liters of water to 30 liters of the solution, what is the new concentration?

Original salt is $0.2\cdot 30 = 6$ liters. New volume is $40$, so concentration is $6/40 = 15\%$.

Example 3: Percent Change

A price increases by 20% and then decreases by 20%. The net change is $1.2\cdot 0.8=0.96$, so a 4% decrease overall.

Strategy Checklist

• Define variables before writing equations.
• Keep units consistent throughout.
• For rates, use a table of rate, time, and work.
• For mixtures, track the amount of solute.

Common Pitfalls

• Mixing up rate and time when adding.
• Forgetting that averages are weighted by amounts, not just counts.
• Treating percent changes as additive.

Practice Problems