GMAT Mixtures, Interest and Profit Problems

Mixture, interest and profit problems turn a plain percent or ratio into a word problem built around blending liquids, growing an investment, or pricing an item for resale. All three types rest on the arithmetic and elementary algebra GMAC lists for Quantitative Reasoning, a section built from 21 Problem Solving questions in 45 minutes with no calculator allowed. The arithmetic stays elementary, so the difficulty lives in the setup rather than in the computation. Once you translate the words into one equation, the numbers fall out in a few steps.

This guide covers mixture and alligation setups, simple and compound interest formulas, profit and markup percent problems, and the weighted-average shortcut behind all of them. Each section gives the formula first, then the reading cue pointing you to it. Three worked examples close the guide, each solved step by step so you see the setup before the arithmetic. Every example here is original, written to match the style of the question types described.

Mixture and Alligation Setups

A mixture problem blends two or more quantities with different rates, prices or concentrations into one combined quantity. The GMAT usually supplies two starting values and a target value, then asks for the missing quantity or ratio. The dressing changes across questions: acid concentrations, alloy metals, coffee blends, ticket prices. The underlying equation stays identical in every version.

Setting Up the Mixture Equation

Every mixture problem reduces to one equation: the total value of the parts equals the total value of the whole. List three rows before touching algebra:

• What you start with: the quantity and rate of the first component.
• What you add or remove: the quantity and rate of the second component joining or leaving the blend.
• What you end with: the combined quantity at the target rate.

Value means quantity times rate, so each row multiplies out to a single number or expression. The table below lays out those three rows for a standard two-part blend.

Set the sum of the first two value cells equal to the combined value cell, then solve for the unknown quantity. This single setup covers blends of liquids, alloys, grades of a product, or scores from two groups of students. Keep units consistent across rows, since a rate per pound paired with a quantity in ounces breaks the equation. When a question removes part of a mixture instead of adding to it, enter the removed amount as a negative quantity and solve the same way.

The Alligation Shortcut

Alligation skips full algebra for two-part mixtures. Three steps produce the quantity ratio:

1. Place the two starting rates at opposite ends of a line.
2. Put the target rate between them, in the middle of the line.
3. Read the quantity ratio off the two differences, pairing each rate with the difference on the far side.

The output is a ratio, not an absolute amount, so you still need one given quantity to finish the job. Reach for alligation when the question asks for a ratio, and use the full equation when it asks for a specific weight or volume.

The ratio of the cheaper quantity to the dearer quantity equals the higher rate minus the target rate, over the target rate minus the lower rate. A mixture priced at 8 dollars per pound, built from parts priced at 6 and 10 dollars per pound, needs the two parts in a 1 to 1 ratio, since 8 dollars sits exactly halfway between 6 and 10. Move the target to 7 dollars and the ratio shifts to 3 to 1 in favor of the cheaper part. The target always sits closer to the rate holding the larger share.

Simple and Compound Interest

Interest problems test growth of a balance over time under two different rules. Simple interest grows by a fixed amount every period. Compound interest grows by a fixed percent of a balance which has already grown. Scan the question for the word simple or compound before picking a formula, because the two results diverge more with every added period.

Simple Interest Formula

Simple interest depends on three inputs: principal, annual rate and time in years. Convert months into a fraction of a year before applying the formula, so 9 months becomes 0.75. The two formulas below return the interest earned and the total balance at the end of the term. Both treat the principal as fixed for the whole period.

Simple interest earns the same dollar amount every period, because each period's interest is calculated on the original principal alone, never on interest already earned. A 5 percent rate on 1,000 dollars pays 50 dollars in year one, 50 dollars in year two, and 50 dollars in every year after. Total interest therefore grows in a straight line with time. This linear growth is the fastest way to spot simple interest inside a table of balances.

Compound Interest Formula

Compound interest earns interest on interest, so the balance grows faster every period. The exponent counts compounding periods, not years, whenever the two differ. The formula returns the final amount first, so subtract the principal to isolate the interest earned. The gap over simple interest starts small and widens with each additional period.

GMAT exponents on these questions stay small, usually 2 or 3 periods, so expanding by hand takes seconds. For 2 periods, (1 + r/100) squared equals 1 plus 2r/100 plus (r/100) squared, and the final term is the extra amount compounding adds over simple interest. Use the expansion when answer choices sit close together and rounding would blur them.

Compounding More Than Once a Year

When interest compounds more than once a year, add a compounding count n to the formula: Amount equals Principal times (1 + Rate divided by n divided by 100) raised to n times Time. Here n is the number of compounding periods per year, such as 2 for semiannual, 4 for quarterly or 12 for monthly. Dividing the rate by n gives the rate per period, and multiplying the time by n gives the number of periods. A 12 percent annual rate compounded semiannually for one year therefore behaves as 6 percent applied twice.

Profit, Loss and Markup Percent

Profit and loss problems compare a cost price to a selling price, then ask for a percent gain, a percent loss, or the price needed to hit a percent target. Cost price is what the seller paid, and selling price is what the buyer paid. A discount applies to a list price, which sits above the selling price whenever a discount is offered. Sort these three prices before writing any equation, since GMAT wording moves between them inside a single question.

Profit and Loss Percent

Profit percent and loss percent both measure the gap between cost price and selling price as a share of cost. Profit percent returns the gain per 100 dollars of cost, and loss percent returns the shortfall per 100 dollars of cost. Only one of the two applies to a given sale, decided by which price is higher. The table below pairs each formula with the condition triggering it.

Both formulas divide by cost price. Profit percent and loss percent always use cost price as the base, never selling price. A 300-dollar sale on a 240-dollar item is a 25 percent profit, since 60 divided by 240 equals 0.25. Switching the base to selling price returns 20 percent, a different figure answering a different question.

Markup vs Margin: Two Different Bases

Markup and margin describe the same dollars of profit against two different bases. Markup measures profit against cost price, and margin measures profit against selling price. Selling price is the larger of the two in any profitable sale, so margin percent always lands below markup percent. The table below sets the two formulas side by side with the base each one uses.

A 25 percent markup and a 25 percent margin never produce the same selling price. Markup adds a percent of cost to cost, while margin sets profit as a percent of the final selling price, pushing the required selling price higher for the same profit percent. Read the question wording closely: markup and margin describe the same profit in dollars, at two different percent scales. Words like markup, marked up and above cost point to the cost base, while margin, profit margin and of revenue point to the selling price base.

Weighted-Average Shortcuts

A weighted-average problem combines two or more groups with different sizes and different averages into one combined average. The setup mirrors a mixture problem: swap liquid quantities for group counts, and swap price per unit for a group average. The combined average always lands between the two group averages, never outside them. The formula below handles the two-group case and extends to more groups by adding matching terms to the top and the bottom.

Multiply each average by its group size, add the products, then divide by the total group size. A plain average of the two group averages is correct only when the two groups are the same size. Any size difference pulls the combined average toward the larger group. Use this pull as a sanity check before you commit to an answer choice.

Weighted Averages as Mixture Problems

Alligation applies to weighted averages the same way it applies to liquid mixtures. Place the two group averages at opposite ends of a line, put the combined average between them, and read the ratio of group sizes off the two differences, exactly as with a two-part liquid mixture. The group whose average sits closer to the combined figure is the larger group. This distance rule alone answers many which-group-is-bigger questions with no arithmetic at all.

Recognizing a weighted-average question as a mixture problem in disguise saves setup time. A combined class average, a blended interest rate across two accounts, or an overall discount rate across two purchase batches all reduce to the same weighted-average formula. Concentration questions hide the same structure, since a solution described as 30 percent acid is a weighted average of pure acid and water. Treat the percent as the rate and the volume as the weight.

Sample Problem Walkthroughs

Three original examples below walk through a mixture problem, a compound interest problem, and a markup versus margin problem, each solved with the formulas above. Every walkthrough states the question first, then the setup, then the arithmetic. Build the setup yourself before reading the solution, because the setup is where most lost points come from. The numbers stay small enough for mental math, matching the no-calculator rule on the section.

Mixture Walkthrough

A shop blends 8 pounds of coffee priced at 6 dollars per pound with an unknown amount of coffee priced at 10 dollars per pound, to make a blend priced at 8 dollars per pound. Find the unknown amount, labeled x. Both starting prices and the target price are given, so the mixture equation applies directly.

Set the value of both parts equal to the value of the combined blend: 8 times 6, plus x times 10, equals (8 plus x) times 8. This expands to 48 plus 10x equals 64 plus 8x. Subtract 8x and 48 from both sides: 2x equals 16, so x equals 8. The shop needs 8 pounds of the 10-dollar coffee, matching the 1 to 1 ratio the alligation shortcut predicts, since 8 dollars sits exactly halfway between 6 and 10.

Compound Interest Walkthrough

An investment of 2,000 dollars earns 10 percent annual interest, compounded yearly, for 2 years. Find the final amount, then compare it with the simple interest result over the same term. Compounding happens once per year here, so the exponent equals the number of years.

Apply the compound interest formula: Amount equals 2,000 times (1 plus 10 divided by 100) raised to the power of 2, which equals 2,000 times 1.21, equals 2,420 dollars. Compound interest earned over the 2 years equals 2,420 minus 2,000, or 420 dollars. Simple interest on the same principal, rate and time would earn only 2,000 times 0.10 times 2, or 400 dollars, 20 dollars less than the compound result. The 20-dollar gap is the second-year interest on the 200 dollars of interest earned in year one.

Markup and Margin Walkthrough

A shop buys a phone for 240 dollars and marks it up 25 percent above cost. Find the selling price, then find the margin percent at this selling price. The word markup fixes cost price as the base for the first step.

Selling price equals cost plus markup: 240 times 1.25 equals 300 dollars. Margin percent uses selling price as the base instead of cost: (300 minus 240) divided by 300, times 100, equals 20 percent. A 25 percent markup produces a 20 percent margin on the same 60-dollar profit, confirming markup and margin never match on the same sale. Answer choices on these questions usually include both figures, so the base you pick decides the point.

Practice these setups with our free Mixtures, Interest & Profit drills, covering three difficulty levels with instant scoring and no signup required. Pair this practice with our fractions, decimals and percents guide for the percent mechanics behind interest and markup, and read our GMAT Quant Reasoning guide for the full section breakdown. Visit our GMAT practice page for every mock and drill, or check your target score with our GMAT score calculator.

Sources

• GMAT Exam Content - mba.com
• GMAT Exam Structure - mba.com