# Logical/Mathematical Relationships Study Guide for the MAT

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## First, an Introduction to the Miller Analogies Test® (MAT®)

There is a brief review of the analogy concept and a broader explanation of how analogies are approached in the MAT® test at the beginning of our MAT® study guide on *Semantic Relationships*. Here’s a link to that information.

## General Information About This Type of Question

One type of mathematical relationship that functions just like an analogy is a proportion. A proportion gives you two entities and then two more that are related in the same way. This type of relationship in a MAT® question operates in the same fashion and can include things like equivalence, patterns, and similar letters and sounds. This study guide can help you spot this type of relationship on your way to finding the correct answer to an analogy.

## Equality

The range of MAT® analogies involving *Logical/Mathematical Relationships* in which the connections between paired items are focused on *Equality* includes comparisons based on equations, numerical fractions, multiples, and negation.

### Equivalence

*Equality* analogies that are based on *Equivalence* can be logical or mathematical in nature. They can require identifying equivalent values in two different systems of measurement, matching important numbers or formulas to their corresponding names, recognizing alternative approaches to representing the same value, or identifying the correct formula for returning a given measurement.

#### Examples

\(80° \text{C}\) : \(176° \text{F}\) :: \(-25° \text{C}\) : (*a.* \(-10° \text{F}\) , *b.* \(-13° \text{F}\) , *c.* \(-27° \text{F}\) , *d.* \(-50° \text{F}\) )

Among the three terms given in this analogy, you can identify an *Equivalence* relationship between the first two: \(80^\circ \text{C}\) is equivalent to \(176^\circ \text{F}\) So, the missing term must be the temperature from the Fahrenheit scale that is equivalent to \(-25^\circ \text{C}\). Since (\(-25^\circ \text{C} \times \frac{9}{5}) + 32 = -13^\circ \text{F}\), the correct answer is “\(-13^\circ \text{F}\).”

\(n\) : \(10\) :: (*a.* \(2n\) , *b.* \(n^{3}\) , *c.* \(3n\) , *d.* \(n^{2}\) ) : \(100\)

Among the given terms here, you can find an *Equivalence* relationship between \(10\) and \(100\), since \(100\) is equal to \(10^{2}\). So, the missing term must be the square of \(n\). The answer is therefore “\(n^{2}\).”

### Fractions

Typically, the construction of *Equality* analogies that are based on *Fractions* will use relationships in which one term is a fraction or a reciprocal of the other. Keep in mind, however, that the fractional values might be expressed in numerical form rather than as literal fractions. So, though they might be explicit in some instances, it is highly possible that in other cases the concept of fractions will merely be implied.

#### Examples

XXV : (*a.* CD , *b.* CL , *c.* VX , *d.* L ) :: D : M

Among the terms given in this analogy, you can identify a *Fractions* relationship between the last two, even though none of the terms is an actual fraction. The Roman numeral D (500) is half of the Roman numeral M (1000). So, the missing term must be what the Roman numeral XXV (25) is half of, which is 50. The correct answer is therefore “L.”

(*a.* \(\frac{1}{2}\\) , *b.* \(\frac{4}{3}\\) , *c.* \(\frac{1}{3}\\) , *d.* 3 ) : \(\frac{15}{5}\\) :: \(\frac{16}{12}\\) : \(\frac{3}{4}\\)

Among the given terms here, you can again find a *Fractions* relationship between the last two. The reciprocal of \(\frac{16}{12}\\) is \(\frac{3}{4}\\) when simplified, and vice-versa.

Hence, the missing term must be the reciprocal of \(\frac{15}{5}\\).

The answer is therefore “\(\frac{5}{15}\)”, or, simplified, “\(\frac{1}{3}\\)”.

### Multiples

In the case of *Equality* analogies that are based on *Multiples*, one term is usually a multiple of the other. Realize, however, that it is not uncommon to find the concept of multiples represented in the form of ratios.

#### Examples

MD : CXCV :: C : (*a.* XIV , *b.* XIII , *c.* XX , *d.* IX )

Among the three given terms seen here, you can identify a *Multiples* relationship between the first and the third. The Roman numeral MD (1500) is 15 times the Roman numeral C (100). So, the missing term must be the number that equals CXCV (195) when multiplied by 15. The answer is therefore “XIII.”

\(\frac{1}{3}\\) : \(\frac{7}{12}\\) :: (*a.* \(8\) , *b.* \(10\) , *c.* \(12\) , *d.* \(16\) ) : \(28\)

Among the given terms in this analogy, you can identify a *Multiples* relationship between the first two. The fraction \(\frac{1}{3}\\) (which has \(\frac{4}{12}\\) as a multiple) and the fraction \(\frac{7}{12}\\) have a ratio of \(4\) to \(7\). So, the missing term and the number \(28\) must also have a ratio of \(4\) to \(7\). The correct answer is therefore \(16\). (Note that both \(16\) and \(28\) are multiples of \(4\), so that when both are divided by \(4\), they yield the desired ratio.)

### Negation

A *negation* is a contradiction or denial. For example, the negation of *x* = 7 is *x* ≠ 7. The symbol “~” or “¬” is used to identify a negation, so the negation of *x* = 7 can be symbolized by “~*x* = 7” or by “¬*x* = 7.” The truthfulness of a negation is the opposite of the truthfulness of its corresponding statement. Hence, if the original statement in a *Negation* analogy is true, its negation will be false. Conversely, if the original statement is false, its negation will be true.

#### Example

3 < 8 : (*a.* 3 ≥ 8 , *b.* 3 ≠ 8 , *c.* 3 > 8 , *d.* 3 ≤ 8 ) :: *A* ⊂ *B* : *A* ⊄ *B*

Among the given parts of this analogy, you can identify a *Negations* relationship between the last two: Since *A* ⊂ *B* indicates that A is a subset of *B*, and *A* ⊄ *B* indicates the *A* is *not* a subset of *B*, the two statements contradict one another. So, the missing part of the analogy must contradict 3 < 8. The correct answer is therefore “3 ≥ 8.”

## Word Play

Do not be surprised to see analogies included in the MAT® test which handle words, not by exploring their meanings, but by treating them as puzzles to be solved. Rather than focusing on vocabulary-related connections, the relationships in these *Word Play* analogies have fun with the spelling, sounds, or letter and sound patterns found within the words themselves.

### Letters

Typically, the construction of *Word Play* analogies based on *Letters* involve the rearrangement of letters to form new words, or even the same word. For example, two possibilities are words that form a different word after losing one or more letters, and palindromes, which are words that remain exactly the same when written backward, such as “racecar.”

#### Same Letters Included

A *Letters* analogy based on the *Same Letters Included* relationship most often consists of one word contained within another word. For example, the word “our” is contained within the word “courage.”

#### Examples

(*a.* uretic , *b.* expense , *c.* surgery , *d.* sergeant ) : URGE :: EXPERTISE : PERT

Among the three given words, you can identify a “Same Letters Included” type of relationship between the last two: One can find the word PERT within the word EXPERTISE. So, the missing term must be a word in which URGE can be found. The answer is “surgery.”

ODE : ROPE :: FLOODED : (*a.* flourished , *b.* property , *c.* sophomore , *d.* roquet )

Among the three words given in this analogy, you can identify the “Same Letters Included” type of relationship between the first and the third: The word ODE can be found within the word FLOODED. So, the missing word must be one in which ROPE is found. The answer is therefore “property.”

#### Same Letters/Different Order

Usually, a *Letters* analogy based on the *Same Letters/Different Order* relationship either involves removing certain letters and reassembling the ones that remain to form a new word, or forming a new word by simply rearranging the original letters as given. This type of word puzzle, where one word is formed by rearranging all the letters from another, is called an *anagram*.

#### Examples

YARD : RAY :: (*a.* rowen , *b.* yardage , *c.* word , *d.* worry ) : ROW

Among the given parts of this analogy, you can identify a *Same Letters/Different Order* relationship between the first two. If you remove the last letter in YARD (the letter D) and reverse the order of the letters that remain, you get the word RAY. So, the missing word must be what you get when you reverse the order of the letters in ROW and then add the letter D. The answer is “word.”

DECIMAL : (*a.* claimed , *b.* decide , *c.* cladode , *d.* numeral ) :: ANGRIER : EARRING

Among the words given here, you can identify a *Same Letters/Different Order* relationship between the last two: ANGRIER and EARRING are anagrams in that you can derive one by rearranging all the letters found in the other. The missing word must therefore be an anagram of DECIMAL. The answer is “claimed.”

### Sounds

Sometimes the two halves of the word pairs in a *Word Play* analogy are connected by some type of sound-related similarity or change, as is the case with homophones and rhymes.

#### Sound Patterns

The word pairs in a *Sounds* analogy that is based on a *Sound Patterns* relationship typically evidence matching or repetitive consonant or vowel sounds.

#### Examples

(*a.* choo-choo , *b.* bric-a-brac , *c.* yum-yum , *d.* hubbub ) : BONBON :: DUM-DUM : TOMTOM

Among the given parts of this analogy, you can identify a *Sound Patterns* relationship between BONBON and TOMTOM, which are both constructed from two identical syllables and pronounced using the short *o* vowel sound. So, like DUM-DUM, the missing part of the analogy must be a word consisting of two identical syllables pronounced using the short *u* vowel sound. The answer is therefore “yum-yum.”

HOBNOB : POPTOP :: HOBO : (*a.* juju , *b.* seesaw , *c.* pigpen , *d.* no-go )

Among the given words here, you can find a *Sound Patterns* relationship between the first two: Both HOBNOB and POPTOP are constructed from two rhyming syllables that are pronounced using the short *o* vowel sound and differ only with respect to the first letter. So, like HOBO, the missing part of the analogy must consist of two rhyming syllables that are pronounced using the long *o* sound and differ only with respect to the first letter. The answer is therefore “no-go.”

#### Homophones

A *Sounds* analogy that is based on a *Homophones* relationship is constructed using words that are pronounced (sound) the same, but have different meanings.

#### Examples

WEATHER : WHETHER :: (*a.* muscle , *b.* missile , *c.* clams , *d.* wither ) : MUSSEL

Among the given words in this analogy, you can identify a *Homophones* relationship between the first two: WEATHER and WHETHER are pronounced virtually the same, but are spelled differently. So, the missing word must be pronounced the same as MUSSEL, though it is spelled differently. The answer is therefore “muscle.”

COURSE : (*a.* curse , *b.* breach , *c.* broach , *d.* screech ) :: COARSE : BREECH

Among the words given here, you can identify a *Homophones* relationship between COURSE and COARSE since they sound the same, but are spelled differently. So, the missing word must be pronounced the same as BREECH. The answer is therefore “breach.”

#### Rhymes

A *Sounds* analogy that is based on a *Rhymes* relationship consists of word pairs that share one or more similar or identical sound units, usually including the final syllable.

#### Examples

(*a.* ring , *b.* tongs , *c.* rough , *d.* flung ) : TONGUE :: FLUFF : RUNG

Among the three given words in this analogy, you can identify a *Rhymes* relationship between TONGUE and RUNG since they sound the same. So, the missing word must sound or end the same as FLUFF. The answer is therefore “rough.”

STARE : THEIR :: SCOFF : (*a.* stuff , *b.* cough , *c.* sheer , *d.* though )

Among the three words given here, you can identify a *Rhymes* relationship between STARE and THEIR since the two words sound the same. So, the missing word must sound or end the same as SCOFF. The answer is “cough.”

#### Other Similar Sounds

The connection between word pairs in a *Sounds* analogy that is based on an *Other Similar Sounds* relationship might stem from: *alliteration*, marked by matching consonant sounds at the beginning of words; *assonance*, which refers to matching vowel sounds; or *consonance*, which is the use of matching consonant sounds.

#### Examples

CATCH : SHUSH :: (*a.* pitch , *b.* grudge , *c.* stamp , *d.* bruise ) : WISH

Among the words given in this analogy, you can identify an * Other Similar Sounds* relationship between the second and last words: SHUSH and WISH exhibit consonance in that the two have matching consonant sounds (i.e., SH). So, the missing word must have a matching consonant sound with CATCH. The answer is therefore “pitch” (since both words have the ch sound).

LOONEY TUNES : (*a.* now and then , *b.* Big Ben , *c.* golden oldie , *d.* Sad Sack ) :: GLOOM AND DOOM : MAD MAX

Among the given parts of this analogy, you can identify an *Other Similar Sounds* relationship between LOONEY TOONS and GLOOM AND DOOM, since they display assonance in the form of matching vowel sounds (*oo*). So, the right choice must have matching (i.e., short *a*) vowel sounds with MAD MAX. The answer is therefore “Sad Sack.”

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