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UMAT test tactics and preparation part 7: worded mathematical problems

Worded mathematical problems on the UMAT exam: translating worded problems into mathematical symbols.

comprehenisve UMAT advice from the team at MedEntry UMAT preparation.

Possibly the most confusing questions on the UMAT exam, worded mathematical problems can actually be solved relatively easily if you understand how the question is written and what the words of the question really mean in mathematical terms.

Prior to solving worded mathematical problems on the UMAT exam, it is essential that you are comfortable with translating words into mathematical symbols. Below is a partial list of words and their mathematical equivalents. You should know these by heart before attempting any of the worded mathematical problems as present in the UMAT exam.










2 plus 2 is 4

2 + 2 = 4


x minus 5 equals 2

x-5 = 2

is the same as

multiplying x by 2 is the same as dividing x by 7

2x = x/7




the sum of y and pi is 20

y + pi = 20


x plus y equals 5

x + y = 5


how many marbles must John add to collection P so that he has 13 marbles

x + P=13


a number is increased by 10%

x + 10%x


the perimeter of the square is 3 more than the area

P = 3+A




x minus y



the difference of x and y is 8

lx-yl = 8


x subtracted from y


less than

the circumference is 5 less than the area

C = A-5


x or * or .


the acceleration is 5 times the velocity

a = 5v


the product of two consecutive integers

x(x+ 1)


x is 125% of y

x = 125%y




the quotient of x and y is 9

x/y = 9


if x is divided by y, the result is 4

x / y = 4


Worded mathematical problems on the UMAT exam: tips on how to go about your ‘translation’.

Although exact steps for solving worded mathematical problems on the UMAT exam cannot be given, the following guidelines will help:

  1. Firstly, choose a variable to stand for the least unknown quantity, and then try to write the other unknown quantities in terms of that variable. For example, suppose that Sue’s age is 5 years less than twice Jane’s age and the sum of their ages is 16. Then Jane’s age would be the least unknown, and we let x = Jane’s age. Expressing Sue’s age in terms of x gives Sue’s age = 2x – 5.
  2. Secondly, write an equation that involves the expressions in Step 1. Most (though not all) worded mathematical problems pivot on the fact that two quantities in the problem are equal. Deciding which two quantities should be set as equal is usually the hardest part in solving a worded mathematical problem. For the example above, we would get (2x – 5) + x = 16.
  3. Finally, solve the equation in Step 2 and interpret the result.


From the example above, we would get by adding the x’s: 3x-5 = 16
Then adding 5 to both sides gives 3x =21
Finally, dividing by 3 gives x = 7
Hence, Jane is 7 years old and Sue is 2x – 5 = 2 * 7 – 5 = 9 years old.


Worded mathematical problems on the UMAT exam: typical UMAT question types and what to look out for when solving them.

Motion problems

Virtually all motion problems involve the formula Distance = Rate * Time, or

 D=R * T

  • Overtake: In this type of problem, one person catches up with or overtakes another person. The key to these problems is that at the moment one person overtakes the other they have traveled the same distance.
  • Opposite Directions:  In this type of problem, two people start at the same point and travel in opposite directions. The key to these problems is that the total distance traveled is the sum of the individual distances traveled.
  • Round Trip: The key to these problems is that the distance going to the destination is the same as the distance returning from a destination.
  • Compass Headings:  In this type of problem, typically two people are travelling in perpendicular directions. The key to these problems is often the Pythagorean Theorem.
  • Circular Motion: In this type of problem, the key is often the arc length formula S = R?, where S is the arc length (or distance traveled), R is the radius of the circle, and ? is the angle.

Work problems

The formula for work problems is Work = Rate * Time, or W = R * T.   The amount of work done is usually 1 unit. Hence, the formula becomes 1 = R * T.

Solving this for R gives R = 1/T

Mixture problems

The key to these problems is that the combined total of the concentrations in the two parts must be the same as the whole mixture.

Coin problems

The key to these problems is to keep the quantity of coins distinct from the value of the coins.

Age problems

Typically, in these problems, we start by letting x be a person’s current age and then the person’s age a years ago will be x – a, and the person’s age a years in the future will be x + a.

Interest problems

These problems are based on the formula:


Often, the key to these problems is that the interest earned from one account plus the interest earned from another account equals the total interest earned:

Total Interest = (Interest from first account) + (Interest from second account) 


For more helpful UMAT tips and UMAT advice please see MedEntry UMAT preparation and keep following our  weekly blog updates on UMAT test tactics and preparation

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