1. Fractions, Decimals, Percents & Ratios

Let's skip the fluff. Fractions, decimals, percents, and ratios are really just four different ways of answering the same question: how big is one number compared to another? Once you can flip between them without thinking, a huge chunk of the ISEE quantitative section turns into free points. But first we need to agree on some vocabulary, because the test writers are very picky about their words. The practice is mixed into the lesson: each question sits right after the idea it needs, and the mixed set at the bottom takes that hint away.

Products & Quotients

Two words show up all over ISEE word problems, and if you mix them up you're toast before you even start computing. The first one is all about multiplying.

Nothing fancy. The only things to watch are the signs and the letters.

A quotient goes the other direction. Instead of combining things, you're asking how many times one amount fits inside another.

Here's a quotient of two fractions, which is exactly where that rule earns its keep.

So the words themselves are the easy part. The order they come in is the trap.

Sort these phrases before the test sorts you. Drag each one onto the division it means.

Now you translate. Say the operation out loud before you compute anything.

Decimals & Percents

Percent literally means "per hundred." So $72\%$ is just a fancy way of writing $72$ out of $100$. That makes converting super mechanical: you're always just multiplying or dividing by $100$.

Two hops, every time. Watch the point go, and watch the zeros fill in when it runs out of digits.

Simple, right? This is exactly where the ISEE sets its traps. They know half the room is going to slide the decimal the wrong number of spots.

Now you guess first, and then the point hops. The two wrong buttons are the two classic slips.

Your turn with the point, both directions.

Fractions convert the same way. You just have to do the division first to get a decimal.

That $.5$ is not a mistake, and it's worth knowing cold.

Here's where the eighths come from. On a grid of $100$ squares, one square is $1\%$, so half a square is half a percent, and an eighth of the grid is twelve and a half squares.

Fractions to percents and back, with a grid or a pizza when there's one to count.

When a problem throws all three forms at you at once, don't try to compare apples to oranges. Pick one language and translate everything into it.

Here are those three landing on one number line, each one translated into a decimal on the way down.

Ordering is always multiple choice, since the choices are the numbers being compared.

Percent Of

In math word problems, "of" almost always means multiply. "Half of $10$" is $\frac{1}{2} \times 10$. Percents work the exact same way.

Drag the handle and watch the part grow with the percent. The marks under the bar tell you what each $10\%$ is worth.

The readout under the bar breaks the percent into tens and ones. That's the mental-math move in this example.

There's more than one good way to break a percent up. Build a few and see which pieces you like.

Now here's a sneaky one. Test-makers use this all the time to reward students who are paying attention.

Here's why, as a block of squares: $8$ rows of $50$ is the same block as $50$ rows of $8$, just turned on its side.

Same squares, so the same answer, and one of the two readings is always easier in your head.

Percent of a number, with the swap whenever it helps.

Sometimes they turn the question around and hand you the part instead of the percent. Don't panic, just translate the sentence word by word. Watch it happen.

Here it is on paper, one line per word.

Part over whole, then times $100$. Try a few.

The last way to turn it around: they give you the part and the percent, and you need the whole.

Here's that backward read as a drag. The part stays put, and you stretch the whole until the part is the right percent of it.

Now find the whole by dividing, no dragging allowed.

Ratios

A ratio is just a quotient that we use to compare two piles of stuff side-by-side.

Written as a fraction, a ratio is a quotient, so every fraction fact you know still works. That's what this rule is really saying.

Try one where the ratio is given and one of the counts is known.

Here's where students get burned. There are two flavors of ratio, and the ISEE loves to give you one flavor and then ask a question about the other.

The other flavor compares a group to everything in the pile.

Same bag of marbles, both flavors.

Write some ratios of your own. Ratios stay multiple choice, since the answer box only reads numbers.

Try the mission below, then flip the switch to see the exact same group as a part-to-whole ratio. Hit reset for a new mission (and a new crowd).

See how the simplest form and the percents never budge when you add batches? Only the actual counts change. That hidden multiplier is exactly what we go hunting for in word problems. First, though, the switch in that widget is a skill all by itself: given one flavor, write the other.

Now the hunt for the multiplier. The giveaway is a part-to-part ratio next to a total.

Here's that school dealt out one batch of $13$ at a time. The number of batches is the multiplier, and the girls are $7$ per batch.

Same trap, different costume: a mixture is a ratio of ingredients, and the total amount is the whole.

One more twist. Instead of the total, they give you the gap between the two groups. The multiplier hides in the gap the same way.

Total or gap, find the multiplier first.

Sometimes two ratios share a group, and you have to link them up through it. The dogs below are in both ratios, but as different numbers, so the cards get scaled until the dogs match.

The same chain, written out.

Now scale the cards yourself. The outside ratio stays locked until the middle numbers agree.

Chain a few on your own.

Proportions

When two quantities grow or shrink together at a fixed rate, they track each other through a scaling multiplier. That's all a proportion is.

That multiplier gets a name of its own, because the ISEE asks for it by name.

Finding $k$ is one division: $k = y \div x$.

Here's a way to picture it. Say every dog at a shelter gets the same number of cookies, $k$. Then the number of cookies is always $k$ times the number of dogs. A lot of students assume "proportional" means "bigger," so keep an eye out for what happens when $k$ drops below $1$.

Did you catch the graph? Every proportional relationship is a straight line that passes right through the origin, and $k$ controls how steep it is. No dogs means no cookies: if $x$ is $0$, then $y$ has to be $0$ too.

Find $k$ from a pair, a story, or a table.

And the recognition version: which of these is proportional at all? Remember the test, $y \div x$ is the same for every pair, and $(0, 0)$ is on the line. This one is multiple choice because the choices are the equations and tables themselves.

Two ratios that are equal make a little equation, and that equation has a name.

The fastest way to solve almost any proportion is to cross-multiply.

Here's the picture behind it. Multiply both sides by $bd$, and each denominator cancels against one factor, leaving the two diagonal products.

Now the page's own proportion, one cross-multiply and one divide.

Solve a few. The unknown moves around, but the move is always the same.

Let's lock this in with a classic scaling problem. Setting up the proportion correctly is $90\%$ of the battle.

The setup is where the points are lost, so here's the rule for it.

Spot the flipped setups. Some of these proportions for the Pokémon problem are fine and some are upside down.

Maps are the same setup with inches and miles.

Drag the second town and watch the ruler and the proportion keep up.

Recipes, prices, maps, and models: one known pair scales to a new one.

Percent Change

Percent change measures how much something moved compared to where it started. Hang on to that word "started," because it's the whole game here.

Here's the classic, with the trap baked right into the numbers.

Stack copies of the change inside each bar. Four fit in the start, five fit in the end, and only one of those is the percent change.

Change over original, times $100$. Up or down, same formula.

Here's a nasty cousin of that trap: one gap, two different percents, depending on which end you measure from.

Same trick every time: call the smaller amount $100$, build the other, and divide the gap by the right one.

When you just want the new amount, skip the middle step and use a multiplier. Going up? Multiply by a little more than $1$. Going down? A little less than $1$.

One change, one multiplier.

Sales, markups, tips, and towns that grow.

Two changes in a row just means two multipliers, each one applied to whatever's there at the time.

Watch the bar shrink and then grow a little back.

Sometimes they hand you the price after the change and ask where it started. Work backward with the same multiplier.

Guess first, then watch the bar model settle it.

Divide by the multiplier. Never add the percent back on.

Now here's the trap that gets almost everyone the first time. The widget below starts on $+50\%$ then $-50\%$. Where do you think you end up?

Here's the same idea with numbers that look like they should cancel and don't.

Up a quarter, then down a fifth, and the bar lands right back on the starting line.

So why does everybody's gut get this wrong? Because the second percent is taken of a different number.

Multipliers, chained. Sale then tax, markup then sale, up then down.

Quantitative Comparison

The ISEE has a question type you won't see on most tests: two columns, and you decide which one is bigger. The four choices never change. (A) Column A is greater, (B) Column B is greater, (C) the two are equal, or (D) the relationship can't be determined from the information given. That last one is the trap door, so check it every single time.

Here are those three values run through both columns. The winner flips, and that flip is the whole answer.

Now you pick the values. Plug in a few and watch for a flip, or for the columns to lock together.

Mixed Practice

Every question so far came right after its idea, which is a hint the real test never gives. Down here the hint is gone. First, the quantitative comparison: two columns, four fixed choices, and any idea from this section fair game.

Next, the mixed set. Every question type on this page except the quantitative comparison goes into one hat, and each click draws a new one at random. Flip the switch to type your answer (fractions like 5/8 and percents like 37.5 are fine); a few types need their answer choices, and they'll say so.

Alright, if you've made it this far, it's time for something with a little heat. These are multi-step problems where you have to chain together everything from this section. Don't rush them. Write down what each ratio is comparing before you do any math.