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]]>I hate to burst your bubble, but this post does not give you the insider trading secrets of the 5 most difficult MAT questions on the exam. Why? Because what is most difficult for you is actually quite personal. As you begin your study regimen, you’ll start to realize which types of questions are most difficult for you and then you can focus your efforts. Even so, here are some examples of what could be trickier MAT questions.

There’s no getting around the fact that the MAT is word-heavy. Some of the words and the concepts they represent are simple, some deceptively so, and some delve into esoteric knowledge. Knowledge of current and historic meanings is important. You can find an in-depth discussion of this in this post about etymology.

For example:

Sinanthropus : Pithecanthropus :: (Peking) : Java

This is an example of needing to know the root words and prefixes. Anthropus indicates that both terms refer to humans and you can use your knowledge of prefixes such as sino- (Chinese) and that Peking was a city in China.

We have had quite a few blog posts on vocabulary. Again, not surprising, given the nature of the exam. Rather than slogging through a series of practice analogies that seem meaningless if you don’t know, take some time to focus on learning broad-spectrum vocabulary words. Take a look at this post about vocabulary to get tips on learning new words.

For example:

Napoleon : Pergola :: (Baker) : Carpenter

This is an example of alternate meanings. Napoleon was a French general but it is also a type of baked good.

Association analogies are the largest group represented on the exam. This means that they will be easy to recognize but don’t get overconfident and blow them off. You can take a look at this post on the analogy subcategory of associations.

For example:

Parrot : (Beak) :: Imaginary : Fable

This analogy shows that one term is a characteristic of the other.

(shudder) We come to my personal challenge–math questions on the MAT. We have several excellent blog posts breaking down analogies that represent mathematical or numerical relationships that I wish I’d had before I took my test.

For example:

4 : 64 :: 5 : (125)

This example uses exponential powers to indicate the relationship. 4 cubed is 64 and 5 cubed is 125.

OK, this is not an actual question subcategory on the test. But it is a way of thinking about difficult questions that may ease your way through difficult questions. Since we know there are 20 experimental questions on the MAT, try to think that the one or two unsolvable or incomprehensible questions that trip you up are experimental. That way you don’t spend too much time on one question. Take a look at this post on experimental questions for more ideas.

I may have not given you the inside scoop on individual analogies but you now have advice to target your 5 most difficult MAT questions. As you study, work on identifying the relationships and focus on the areas most difficult for you to create for yourself a personalized, targeted study experience.

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]]>In the past few posts, we’ve built up a nice little inventory of *word game analogies*: special subtypes of question that you might encounter on the MAT. (Not sure what I mean by “word game analogy?” See this post for starters.) We’ve looked at some examples involving __spelling__, as well as a few involving pronunciation.

Now we’re headed into the gray area of etymology (word origins) and usage. In a sense, these are *semantic* properties, because the way a word is used affects how it is understood. (If people started shouting “ice cream!” every time they stubbed a toe or lost a round of *Overwatch*, it wouldn’t be long before “ice cream” was understood as an expletive.) Distinctions of usage and etymology on the MAT are what we might call “punch-up” traits – that is, they enhance the difficulty of otherwise easy questions.

The MAT seldom tests words in foreign languages *per se*. However, loan words like *sushi *or *macho* will appear fairly often. Even more common — on the MAT and in everyday vocabulary — are words that have been in the language so long that we don’t usually think of their origins. This next example consists entirely of such terms.

*spy : clandestine :: sailor : ________*

*(A) seafaring
(B) nautical
(C) nauseous
(D)*

Let’s assume that the word *clandestine *(“deliberately secret or secretive”) doesn’t throw us off. (If that assumption’s inaccurate, it’s time to write up a new vocab flash card!) Focusing on *semantics* first, we might say that “a *spy *is engaged in *clandestine* activities.” That’s a pretty straightforward relationship. And a *sailor *is engaged in … well, either *seafaring* (A) or *nautical* (B) would fit here on the basis of meaning alone. We’ll break the tie in a moment, but first let’s rule out those other answers. A sailor – we hope – isn’t habitually engaged in nauseous (C) activities, so we can strike that from the list. *Seasick** *(D) takes us to the same place as *nauseous*.

All of our answer choices have to do with the sea. Two of them, however, contain Latin roots for sailing and ships (cf. *nauta*,* nausia*); the other two use the Germanic word “sea” (cf. modern German *See* or Dutch *zee*). This is all we need to clear up the ambiguity between *seafaring *(A) and *nautical *(B) – our winner is (B). Here’s one way we might express the analogy.

*Clandestine *is a Latin term describing a *spy*’s activities.

*Nautical* is a Latin term describing a *sailor*’s activities.

Etymology on the MAT isn’t always a tiebreaker like this. It can also serve the simpler (and less frustrating) purpose of helping you to break down unfamiliar words. The MAT Study Guide (p. 13) models this use in a question about *Sinanthropus*, an extinct hominid whose fossils were discovered in the 1920s. If you’re not an anthropology or archaeology major, it’s quite possible that a word like *Sinanthropus* doesn’t mean anything to you. But as the study guide shows, there are still ways to make an educated guess.

Usage can be just as important as etymology on the MAT. The knowledge of how, where, and when a word is used can be the missing piece in an otherwise baffling analogy. Consider this example, which reflects the deceptively simple nature of many MAT analogies.

*vest : waistcoat :: _________ : trousers*

*(A) shoes
*

Nouns describing clothing – like most concrete nouns – are rich in potential *semantic* relationships: part/whole, inner/outer, large/small. *Shoes* go over *socks*, a *sleeve *is part of a *shirt*, etc. But perhaps because these nouns are so ordinary, there are often important differences in *usage* from country to country: a *jumper* in the UK is a *sweater* in the US, for example.

The analogy above likewise involves a distinction of usage. The pairs of terms mean the same thing in different varieties of English. In the English of the United Kingdom, a *vest* is a type of undershirt, and a *waistcoat* is a sleeveless garment worn over a shirt. In the United States, that same sleeveless garment is called a *vest*. Likewise, in UK English, “*pants*” is short for “underpants,” and *trousers* are the outer garment worn on the legs (e.g., khakis or jeans). In US English, that same article of clothing is called *pants*. Putting that all together, we get the following analogy:

The garment called a *vest* in the US is called a *waistcoat* in UK usage.

The garments called *pants *(D) in the US are called *trousers* in UK usage.

At this point, you may well be wondering: does the MAT really test this sort of trivia? Trucks versus lorries? Elevators versus lifts? These distinctions seem like material for family-friendly stand-up comedy, but surely not for a graduate admissions test … right? Alas, no. In the officially published practice questions, the MAT has shown that seemingly incidental bits of cultural knowledge are fair game for analogy-building. If botanical terms (Study Guide p. 8) and pastries (p. 12) can make the list, then quirks of English usage are fair game, too.

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]]>In the last post in this series, we took a brief tour of an unusual but important type of MAT analogy: questions based on a word’s spelling, rather than its meaning. A similar class of analogies are built on the basis of pronunciation. As noted in their official publications, the MAT test makers will sometimes use words with counterintuitive meanings, like “inflammable,” or with multiple meanings (e.g. “sanction”). Similarly, words with counterintuitive pronunciations – or multiple pronunciations, for that matter – can be used to crank up the challenge level. Like most types of question on the exam, MAT pronunciation analogies will vary in difficulty from “easy” to “extremely hard.” In the following discussion, I’ll use examples of moderate difficulty so that you can see the key principles at work.

English is full of words with silent letters: French borrowings, nautical terms, and so on. These words are exactly the sort of cultural shibboleth – like the Greek alphabet or the names of the U.S. presidents – that the MAT loves to include in its analogies. In fact, it’s quite possible that you’ll face an MAT pronunciation problem that gets its difficulty *entirely *from silent letters. Here’s what that might look like:

*Worcester : Versailles :: Gloucester : _________*

*(A) **Illinois
*

All seven of these words – stem and answer choices – are place names. *Worcester *is a city in the English Midlands, and Gloucester is a city in South West England. *Versailles *is the site of the famous chateau of Louis XIV of France. Readers of Shakespeare might also recognize some characters from the history plays. The Earl of *Worcester* spearheads the rebellion in *1 Henry IV*, and in *Richard III*, the eponymous evil king starts out as the Duke of *Gloucester*.

A good starting point might be: “what do these places have in common?” *Worcester *and *Gloucester* are in the same country, which seems promising, but *Versailles *is in France, and none of the answer choices are on the European continent. *Worcester *and* Gloucester *are both cities – again, a potentially useful lead – but *Versailles* is a palace (or the city nearest that palace), and the answer choices are all North American states and provinces. (MAT pronunciation analogies, like *all* MAT analogies, have a tendency to favor U.S.-centric subject matter.)

So neither size nor location helps us out much. Stretching a bit, we might try to fit *Worcester* and *Gloucester* together by the fact that they’re “Shakespearean bad guys named after places.” But there’s no such thing as an Earl of Versailles or a Duke of Versailles: the palace was built by, and for, the king and his court. And you won’t find a stage direction like “Exeunt Jalisco and Saskatchewan” anywhere in the Bard’s works.

However, if we step back from *meaning* and focus on *pronunciation*, a viable pattern emerges. *Worcester *(pronounced /ˈwʊstər/ or “Wooster”) and *Gloucester *(/ˈɡlɒstər/, to rhyme with “foster”) have silent letters in the middle. *Versailles* (English pronunciation /vərˈsaɪ/, to rhyme with “goodbye”) has a silent *s* at the end; only (A) *Illinois *(/ˌɪlᵻˈnɔɪ/) shares that trait. So our answer is (A), and our analogy goes something like this:

*Worcester* is to *Gloucester* as *Versailles *is to *Illinois*. The first two have silent letters in the middle; the latter two have silent letters at the end.

This next analogy is a bit more challenging. Take your best shot and then read on.

*produce : permit :: profile : _________ *

*(A) **progress
*

As usual, the first thing to ask is “what do these words have in common?” Well, the three stem words all start with the letter P*, *but that doesn’t narrow things down: all of the answer choices start with P too. There’s no clear pattern of *meaning* here either. In fact, the sheer repetitiousness should put us on our guard for wordplay. Weird repetitions don’t *guarantee *that we’re dealing with a word game, but MAT pronunciation problems (and other wordplay problems) often have a singsong, riddle-like quality.

What holds these words together is their pattern of *syllable stresses*. In *produce *and *permit*, the stress changes based on the part of speech. As a noun, meaning “fruits, vegetables, and other farm products,” * produce* has its stress on the first syllable; in the verb form (“to make, yield, or create”), the stress comes on the second syllable:

So much for terms 1 and 2. But how does that observation help with terms 3 and (blank) 4? Like the first two terms, *profile* represents multiple parts of speech. To *profile *(v.) someone – say, an artist you’re interviewing in a magazine – is to create a *profile* (n.) of them. But unlike *produce *and *permit*, *profile* has the same stress pattern in either form. That pattern – a similarity and a difference – puts us on alert for an answer with the same properties.

*Progress *(answer A) and *project *(answer B) are no good because they *do* change syllable stress depending on part of speech. A *profiterole* (known as a “cream puff” in US English) is a noun only; you can’t *profiterole* something (more’s the pity!). That leaves (C) *program* as the only word that

- exists as multiple parts of speech
- always has the same syllable stress pattern.

Once it’s grasped and explained, this type of analogy can often feel like a “gotcha” or a cheap shot. It’s tough stuff, I know! But bear in mind that we’ve been focusing on the most “puzzle-like” part of the MAT. Get familiar with these patterns, learn to anticipate the MAT’s traps and tricks, and you’ll gradually rob the exam of its power to surprise you.

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]]>In a previous post, I pointed out a few ways that the MAT might construct an analogy based solely on spelling. Here’s a more detailed list of MAT spelling games, populated with fresh examples in each category.

- The analogy might be based on
*anagrams*, words that are rearrangements of the same set of letters (e.g.*lane : lean*). *Semordnilaps*are a special case of*anagrams*in which the spellings of two words are not just rearrangements of each other, but mirror images. A well-known example is*stressed : desserts*. (The term*semordnilap*, if you’re wondering, comes from ‘palindromes’ spelled backwards – and yes, it’s a real word!)- Sometimes, each pair of words will have a
*sequence of letters*in common. Usually, the initial letters are the important ones, as in*reach : realm*, but you might also see an example in which the matching sequence occurs in the middle or at the end. - Finally, you may come across an analogy in which the words have corresponding
*sets of letters*, though not necessarily in the same order. In*devious : audio*, for example, the defining trait is the presence of four distinct vowels.

So how will those patterns appear on test day? Unfortunately, there’s no reliable cue to tell you when you’re dealing with a spelling- or pronunciation-based analogy, rather than the more common semantic kind. That said, there are a few circumstantial hints to look for, as I’ll show in the following two examples. You’ll see how a few basic patterns can combine to form a variety of MAT spelling analogies, from the elementary to the esoteric.

First, something nice and simple:

*goal : gaol : bolt : _________*

*(A) **bold
*

Here, there’s an interesting but vague *semantic *relationship between the first two terms, *goal *and* gaol *(a UK spelling of *jail*). Playing around with the meanings of the words, we might come up with a sentence like this:

“Brought up shoplifting charges, Harry made it his *goal *to stay out of *gaol*.”

But as rich as the potential for wordplay here is, we *can’t* say that *goal *and *gaol *are *opposites*. Moreover, it’s hard to see how the meaning of *bolt *and any of the answer choices could be worked into that pattern. For that matter, these words are oddly short and simple to be part of a vocabulary question. A “too good to be true” word list like this is often — but not always — a sign of an MAT spelling game.

When there’s no clear *semantic* cue to follow, it’s time to look instead at the *forms* of the words, starting with the spelling. As it happens, we can form a nice, succinct analogy that relates the spellings of pairs (1, 2) and (3, 4):

*Gaol *is an anagram of *goal* with the middle letters reversed. *Blot *is an anagram of *bolt *with the same property.

This analogy points unequivocally to answer (C), *blot*, as the answer.

Here’s another practice problem to prime you in your recognition of MAT spelling patterns.

*rhythm : _________ :: sequoia : euphoria*

*(A) pyx
*

Although you might feel a sense of *euphoria* among the giant redwoods of California, there’s no strong *semantic* relationship between terms 3 and 4 here. Instead, this question is built around spelling — specifically, around an unusual property of these three words. *Sequoia *and *euphoria *each include all five of the traditional vowels: AEIOU. *Rhythm*, in contrast, has no vowel letters – only the “semivowel” letter Y. That makes (A), *pyx*, the best match here.

Notice also that one of the incorrect options, (D) *adagio*, was thrown in simply because it has to do with music. An over-hasty test taker might choose it just because it seems to fit with *rhythm*. Another incorrect choice, (B) *eudaimonia*, was included mainly because of its visual resemblance to *euphoria.* (The two words refer to different kinds of happiness.) You’ll want to watch out for similar “too easy” answers on the real MAT.

Whenever you suspect that an MAT question involves wordplay, step back and remember: *most questions test relationships of meaning, not form. *As this next example illustrates, there will often be (wrong) answers that seem plausible if you’re already looking for a word game analogy. If you go in with a meaning-first approach, you won’t fall into these traps.

*ichthyosaur : giraffe :: pterodactyl : _________*

*(A) **salmon
*

When we’re thinking in “word game” mode, *pterodactyl : ptarmigan *seems like a reasonable pair, because of the initial “pt-” in both. But there’s no corresponding pattern in *ichthyosaur : giraffe* – in fact, those two words have very little in common spelling- or pronunciation-wise. To solve this analogy, we need to stick to the meanings of the words in front of us.

All three of the given terms refer to animals, so we can probably rule out (D) *cactus* at a glance. But* e*ven with that restriction, there are a bunch of analogies we could form. The trouble is that most of these analogies leave us with multiple correct answers:

*Ichthyosaurs and pterodactyls are extinct; the giraffe and the {flying fox, ptarmigan, salmon} are extant.*

This is a warning sign! If an analogy seems like it would fit with more than one answer, the relationships involved are too vague. Instead, we need to be as specific as possible about what unites the terms:

*Ichthyosaurs and pterodactyls are extinct reptiles; the giraffe and the flying fox (and *only

By considering multiple types of semantic relationships, we can narrow this question down to a single answer: *flying fox *(B). The resulting analogy sorts the terms into two distinct, symmetrical categories, with no room for other answers to take the place of answer (B). That kind of clear-cut analogy is our goal in every MAT problem, and I’ll have more to say about it in the next post.

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]]>Some of the trickiest questions on the MAT are what we might call *word game analogies*. Although you won’t find this phrase in a Pearson publication, these are a distinct and important type of question that relies on the spelling, pronunciation, and usage of words rather than their meaning. These questions are, in essence, little puzzles that test your creativity in identifying different types of relationships. They perform a number of functions within the larger test structure:

- Word game analogies can test your knowledge of a word in ways that extend beyond its dictionary definition. (Do you know how it’s pronounced? its language of origin?)
- Because they play on relationships of form, rather than of meaning, word game analogies can be time-consuming, draining away precious minutes from later questions.
- Even more deviously, word game analogies can make you second-guess your approach to other, easier questions, potentially causing you to overlook simple relationships of meaning.

It’s important to note that this type of analogy is relatively rare on the MAT. More often, MAT questions will depend on *semantic* relationships, or relationships of meaning. Bertrand provides a great overview of these questions in his series on MAT analogy subcategories. In the simplest case, the terms in an MAT analogy will be synonyms or antonyms; more complicated relationships include “part/whole” or “action/agent.” It’s safe to say that, for most questions on the exam, the *meanings *of words will be more relevant than their spoken or written *form*.

Because word game analogies are rarer than the semantic variety, it’s essential that you *look for “word game” relationships as a last resort – after you’ve considered more conventional analogy-building strategies. *The makers of the MAT are well aware that test takers tend to spot “word game” patterns even when they’re not important. To that end, the exam will include some questions that

*loaf : laze : _________ : young pig*

*(A)* *load*

*(B) foal*

*(C) strophe
*

If you approach this question expecting some kind of language trick, you encounter a veritable smokescreen of distractions. Answer (A), *load*, begins with the same three letters as *loaf*, and answer (B), *foal*, is an anagram. Answer (C), *strophe*, may look like it *should* rhyme with *loaf*, but the final *e *isn’t actually silent: the word is pronounced /stroʊfi/, to rhyme with “Sophie.”

So three of our four answers have some kind of non-semantic relationship to *loaf*. (*Loaf*, as The Four Aces once sang, really *is* a many-splendored thing.) But none of those relationships serve as a basis for a complete analogy, because *laze* doesn’t have a corresponding relationship to the phrase *young pig*. The analogy here is strictly semantic: to *loaf* is to *laze*, and a *shoat* (answer D) is a *young pig*.

Now that we’ve skirted that potential pitfall, what do I actually mean by “word game analogies?” There are several different kinds, but I think it’s easiest to group them into three main categories. In a word game analogy, the relationship will be based on how the words are *written*, how they’re *spoken*, or when and how they’re *used*.

- Some analogies will be based on the
**written form**of the words. The “missing” word may be an*anagram*of one of the words provided (*tape : pate*), or have some pattern of*initial or final letters*in common (*peal : pear*). - Others will be based on
**pronunciation**.*Rhyme*is the most obvious relationship here, but two words might also have some other phonological property in common, such as the number of*syllables*(*nu : strengths*) or the pattern of*syllable stresses*(*carnival : Washington*). - Finally, some word-game analogies reflect the
**context**in which a word is used. Even within English, there are many examples where usage varies by*country*or*region*(*truck : lorry*). And there are many, many cases (thanks, Norman Invasion!) in which two words with the same basic meaning will have quite different etymologies (*dog : canine*).

In the next few posts, I’ll dive into each of these categories, providing examples of how they might appear on the test.

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]]>

Analogies can be classified based on the type of relationship they present. There are four main relationship types tested on the MAT: semantic, classification, association, and logical/mathematical. For in-depth coverage, I suggest reading through the official MAT study guide.

Semantic analogies create relationships based on the literal definition of the terms given. The terms are linguistically connected and may represent synonyms, antonyms, or other types of definition based connections:

Vacillate : Waver (Both words mean to be indecisive.)

Breeze : Squall (A squall is a violent wind; these represent different levels of intensity for the same concept.)

Amenable : Pugnacious (Amenable is friendly; in contrast, pugnacious refers to a standoffish, disagreeable person.)

Classification relationships are based on either both terms belonging to the same category, or else one term is contained within another:

Emotion : Joy (Joy belongs in the class or category of emotions.)

Wolf : Jackal (Both wolves and jackals belong in the same genus — Canis.)

Girder : Building (A girder is part of a building.)

This relationship features logical or mathematical equations, numerical fractions, multiples, negation, or letter and sound patterns.

1/2 : 1/20 (1/2 is 10 times as much as 1/20.)

7 : 11 (7 and 11 are consecutive prime numbers.)

Rant : Slant (Rant and slant rhyme.)

Association relationships deal with two distinct but related ideas. In a sense, this is a catch all category featuring any relationship that would not fit neatly into the other three.

Snake : Scales (A snake has scales.)

Hurricane : Damage (A hurricane typically produces damage.)

Deaf : Hearing (To be deaf is to lack hearing.)

This is just a quick overview of the types of analogy relationships featured on the MAT. For more information, see the MAT Official Guide. I would also suggest looking at some of our relationship-specific blogs, such as MAT semantic analogies. You can also try your hand at over 100 practice questions using the Miller Analogies flashcard app (free).

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]]>Previously, we covered semantic and classification analogies on the MAT. Now we are going to move into MAT association analogies. Association is the largest category on the Miller Analogies Test; it’s also where you’ll find the most diverse kinds of connections between terms. Because of this, association is capable of producing some really challenging analogies.

“Association” is an interesting term because, in reality, all analogies depend on *associations; *if the terms in an analogy didn’t relate to each other through some sort of association, you wouldn’t be able to form an analogy at all. You would just have four unrelated terms.

In a way, the association category is really the “catch-all” category. The other categories on the MAT focus on very specific associations between terms: classification, semantics, logical/mathematical. But these are just three ways in which ideas can relate to each other.

If you think about it, you’ll quickly realize that words can relate to each other in dozens, if not hundreds, of ways. Association is how the MAT categorizes analogies that wouldn’t fit neatly into the other categories.

For example–

**Allegory : Moral : Apocryphal : False**

Allegories often tell a story which has a *moral* quality. The main quality of apocryphal stories is that they are largely, or entirely, *false*. This type of connection is outside the other categories and is characteristic of association.

We’re going to dive into some examples of association analogies sub-types, but I would highly recommend also looking at the MAT official study guide for a more expansive list of possible association sub-types.

**Object/Characteristic – one term is a characteristic, source, or location of another.**

*One term is an attribute of another:*

Horse : Mane

*One term as an attribute the other lacks:*

Prisoner : Freedom

*One term is the source of another:*

Practice : Proficiency

*One term gives the location or setting of another:*

Tigris : Turkey

**Order – The terms are in a sequential or reciprocal relationship to one another. **

*The terms have a time or sequential relationship but do not cause one another:*

4 : 6 (sequential composite numbers)

*The terms have a reciprocal relationship so that one concept cannot exist without the other:*

Father : Child

(*This one needs more elaboration: A father must have a child to be a “father,” a child must have a father (biologically) to exist*. *They are reciprocal*.)

**One word is a grammatical transformation of the other:**

Curriculum : Curricula

I’m going to end with some practice questions, but I highly recommend anyone in the midst of studying go to the official MAT study guide and really get to know association analogies in greater depth.

- Snake : Porcupine :: (a. scales b. fangs c. rattle d. coils) : Quills
- Maladroit : (a. simulacrum b. conscious c. dexterity d. generosity) :: Servile : Pride
- Data : Datum : Algae : (a. alga b. algaes c. algaes d. augur)

For more practice questions, we have also released free MAT flashcards which feature 160 example analogies that are free to study.

Answers to the above question:

1. (a)

2. (c)

3. (a)

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]]>MAT logical/mathematical analogies are the last category to cover on the Miller Analogies Test. This category contains math, non-semantic wordplay (think palindromes), and a variety of analogies that have a “riddle” or a “brainteaser” feel to them.

The majority of students who take the MAT tend to be verbally focused. Many of you are well-versed in both math and verbal. However, some of you may be the type of verbally gifted student who has avoided math like the plague.

While you don’t need to master high-level math to excel here, the following basic concepts are pivotal to success:

**Fractions and Percents**– You must know how to convert one to the other and vice versa.**Composites and Primes**– Know the differences.**Primes**– Much more important than the composite numbers, you should get familiar with prime numbers. If possible, memorize the order of primes from 1-100. Primes are a favorite test of pattern recognition on the MAT.**Squaring and Cubing**– You should know how to do these operations. You won’t have to calculate big numbers, but you need to be able to see, for example, 10^3 and 1000 are equivalent.**Negative and Fractional Exponents**– The MAT simply wants to see if you can spot the equivalence between two expressions: do you know that 10^-2 = 1/10^2 = 1/100? If you learn how to express a number with a negative or fractional exponent, that will be enough.**Positive and Negative radicals**(Roots) – Again, no big calculations, just equivalencies. I put emphasis on the “radical” sign because the MAT loves to use radicals.**Geometric Shapes**– The MAT likes to relate polygons through their number of sides. To answer such questions, you need to learn the common polygons: triangles, rectangles, pentagons, etc. You don’t necessarily need to learn the names of 12+ sided figures, but instant recall of the features of something like an octagon is necessary.**Geometric Formulas**– You need to know how to calculate area and perimeter for triangles, rectangles, circles, and spheres.

If any of this is unfamiliar to you, it’s very easy to brush up on these concepts through Khan Academy. There may be other math concepts that appear, but the above will cover nearly any question thrown at you. On to examples–

These items will contain any of the following: logical or mathematical equations, numerical fractions, multiples, negation, or letter and sound patterns.

**1. One term is a fraction or multiple of another.**

12 : 144 144 is the square of 12.

4/20 : 1/5 4/20 can be simplified to 1/5.

**2. The terms are related through some non-semantic similarity or change: rhyming, homophones, letter reversal, or other wordplay.**

Live : Evil “Live” in reverse spells “evil.”

Listen : Silent “Silent” is an anagram for “listen.”

- 125 : 64 :: 5 : (a. 1/2 b. 4 c. 1 d. 3)
- 3 : 5 :: (a. 13 b. 17 c. 11 d. 4) : 13
- (a. stuff b. though c. thought d. sow) : Tough :: Brow : Bow

If you have not been to the MAT official Guide or looked through the other MAT analogy blogs, I suggest checking them out for more insight into the different types of analogies.

Answers–

- b
- c
- b

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]]>The post How the MAT Tests Math, Part 4: Geometry appeared first on Magoosh MAT Blog.

]]>The simplest MAT geometry questions, in my opinion, are the ones that present you with a pair of formulas and a pair of terms that describe those formulas. Consider the following setup:

Rectangle : w × h :: Circle : _________

*(A)* 2πr

*(B)* r^{2}

*(C)* (πr)^{2}

*(D)* πr^{2}

The expression *w × h* (width times height) is the formula for the area of a rectangle. So we need to look for an expression that gives the area of a circle. The correct formula is **(D)**, *πr ^{2}*, but note that the circumference formula (

Other questions will ask you about *notation*: the symbols conventionally used to indicate various geometric concepts.

→ : Ray :: : _________

*(A)* Line

*(B)* Line Segment

*(C)* Chord

*(D)* Tangent

In geometry, there are three basic types of one-dimensional figures – shapes with length, but no width or depth. *Lines* have no endpoint; they extend infinitely in both directions. A *ray* has a single endpoint; a *line segment* has two. When drawing these shapes, it’s customary to use an arrow to indicate the ends that extend infinitely, but not those that are bounded by an endpoint.

With that in mind, here’s our analogy:

“A one-sided arrow is used to denote a *ray*; a two-sided arrow is used to denote a *line*.”

Consequently, our answer choice is **(A)**. Again, typically of an MAT geometry question, the incorrect answer choices are also “geometry-flavored,” adding to the problem’s difficulty. A *chord* is a special type of line segment that intersects a circle at both endpoints. A *tangent* is a line that touches a curve (e.g., a circle) at exactly one point.

However, MAT geometry questions won’t always spell out the formula you need to apply; sometimes, you’ll need to recognize the opportunity to use a formula in the first place. In this next problem, for example, it’s evident that there’s some kind of relationship between a *radius* and a *circumference*.

3 : Radius :: _________ : Circumference

*(A)* 6

*(B)* 6π

*(C)* 9

*(D)* 9π

Here, the key is the formula *C =* *2πr* (the circumference of a circle is 2π times its radius), which featured as one of our answer choices in a previous problem. Using this expression, we can figure out the analogy:

“_________ is 2π times 3, just as a circle’s *circumference* is 2π times its *radius*.”

The answer is **(B)**, since 2π × 3 = *6π*.

Here’s another, more challenging question in a similar vein:

Cylinder: 8 :: Cone : ________

*(A)* 2

*(B)* 2 2/3

*(C)* 3

*(D)* 3 3/4

We might begin a problem like this by asking if the number 8 has any special relationship to cylinders in general: “a cylinder has eight _________” or “there are eight _________ in a cylinder.” This, as you’d soon discover, is a dead end. Instead, we can try relating *cylinder* to *cone*. What properties do cylinders and cones have in common?

Well, they’re both solids, which means that each has a *surface area* and a *volume*. Of those two properties, it’s *volume* that holds the clue to this question. Here are the volume formulas for a cylinder and a cone of radius *r* and height *h*:

V_{cylinder} = πr^{2}h

V_{cone} = (1/3)πr^{2}h

Note that the only difference between the two is that factor of 1/3. A cone of a given radius and height is 1/3 the volume of the corresponding cylinder. Now let’s capture that relationship in an analogy. It might go something like this:

“_________ is 1/3 of 8, just as a *cone* is 1/3 of a *cylinder*.”

8 × 1/3 = 2 2/3, so answer **(B)** is our winner.

As with algebra, the geometry that appears on the MAT is largely high-school fare, meaning you’ll need to refresh your memory, but you won’t need to learn it from scratch. Properties of circles and triangles are common, and you may see a question about other simple shapes (e.g. quadrilaterals) or basic solids (e.g. cylinders, prisms, spheres). It’s possible, but unlikely, that you’ll see questions about complicated figures like nonagons or icosahedrons. If you’re trying to brush up in advance of the test, I recommend focusing on the material in our GRE Math Formulas eBook, as this corresponds to the lion’s share of MAT geometry questions.

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]]>The post How the MAT Tests Math, Part 3: Algebra appeared first on Magoosh MAT Blog.

]]>In the past two posts, we’ve looked at the fundamentals of MAT math, covering both arithmetic and some basic number theory. This post picks back up with algebra, another important part of the Mathematics content area. As you might expect, MAT algebra problems can be tricky, since you have limited information to help you along. The good news is that MAT algebra is also “distraction-free,” with no weird charts or word-problem padding to obscure the answer.

Some MAT algebra questions are just stripped-down versions of what you’d see on other standardized tests. The following practice problem, for example, would be turned into a big production on the GRE, with a paragraph of distracting information about factory output or the dimensions of a layer cake. On the MAT, for better or for worse, we get only the bare mathematical details:

x – 2 : 7 :: (x + 1)^{3} : _________

*(A)* 512

*(B)* 729

*(C)* 1000

*(D)* 1331

In this case, the “analogy” is actually an equation, or more accurately, a pair of equations:

If x – 2 = 7, then (x + 1)^{3} = _________

Solving for x is the way to proceed here:

x – 2 = 7

x = 2 + 7

x = 9

Then we plug our newfound value of x into the more complicated expression in slot 3:

(x + 1)^{3} = (9 + 1)^{3}

= (10)^{3}

= 1000

The algebra topics on the MAT are overwhelmingly ones you’d encounter in a standard high-school Algebra I course. You might, for example, have to multiply together a pair of binomials (i.e., “FOIL” them) or factor a quadratic expression. (If those terms make you uneasy, you can find a quick review in the Algebra section of our GRE Math eBook.) But many of the hardest-hitting algebra problem types, such as systems of equations or systems of inequalities, are off-limits for the MAT. The question format simply doesn’t leave room for such problems.

Now, a handful of MAT algebra questions, perhaps one or two per exam, will be purely terminological. Consider the following example:

x^{2}: quadratic :: _________ : quintic

(A) x^{3}

(B) x^{5}

(C) x^{7}

(D) x^{8}

There’s no actual algebra required in this problem. Instead, the question asks us about the *naming conventions* for different powers of a variable. You may already know that an expression with an x^{2} term in it is a quadratic expression. You might even recognize that an expression with an x^{3} term is *cubic*. But what happens for powers larger than three? Here, we fall back on Latin ordinal prefixes: *quartic *(think “quartet”) means “fourth-degree,” and *quintic *(think “quintuplets”) means “fifth-degree.”

The analogy here goes something like this: “Just as x^{2} is a (very simple) *quadratic* expression, x^{5} is a (very simple) *quintic* expression.” In case you’re wondering, the usual terms for higher-degree expressions (or polynomials or functions) are *sextic *for sixth-degree, *septic* for seventh-degree, and *octic* for eighth-degree. Beyond that, even the mathematicians don’t care enough to have a consistent naming scheme.

The upshot is that “MAT algebra,” like “MAT chemistry” or “MAT history,” often boils down to specialized vocabulary. In the cramped multiple-choice format of the exam, there’s not much else they can test.

Notice, by the way, that the above question works even if the variable *x *is never mentioned. A more abstract, slightly more difficult version might read as follows:

2 : quadratic :: _________ : quintic

(A) 3

(B) 5

(C) 7

(D) 8

Here, we can state the governing analogy nice and compactly: “Quadratic expressions include terms of degree 2, but no higher; quintic expressions include terms of degree 5, but no higher.”

Like other posts in this series, the above is a brief sketch of the main ways in which the MAT deploys algebraic terms, concepts, and principles. If you feel that your grasp of those basics is shaky, I again recommend our GRE Math Formula eBook as a starting point for figuring out what you need to review. My goal here, as throughout the series, is to prime you to recognize the *types of relationships* that the MAT favors in its math problems.

With that in mind, we’ll be moving on to geometry in the next post. See you there!

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