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NUMBER WORD PROBLEMS WITH SOLUTIONS

Problem 1 :

The sum of 5 times a number and 8 is 48. Find the number.

Let x be the number.

5x + 8 = 48

Therefore, the number is 8.

Problem 2 :

A number consists of three digits of which the middle one is zero  and the sum of the other digits is 9. The number formed by interchanging the first and third digits is more than the original number by 297. Find the number.

Let x 0 y be the required three digit number.

Given : The sum of the other digits is 9.

y = 9 - x ----(1)

Given : The number formed by interchanging the first and third digits is more than the original number by 297.

y 0 x - x 0 y  = 297

[100(y) + 10(0) + 1(x)] - [100(x) + 10(0) + 1(y)] = 297

(100y + 0 + x) - (100x + 0 + y) = 297

(100y + x) - (100x + y) = 297

100y + x - 100x - y = 297

-99x + 99y = 297

Divide both sides by 99.

-x + y = 3 ----(2)

Substitute y = 9 - x.

-x + 9 - x = 3

-2x + 9 = 3

Substitute x = 3 into (1).

x 0 y = 306

Therefore, the three-digit number is 306.

Problem 3 :

Of two numbers, ⅕ th of a the greater equal to ⅓ rd of the smaller and their sum is 16. Find the numbers.  

Let x and y  be the two numbers such that  x > y .

Given : ⅕ th of a the greater equal to ⅓ rd of the smaller.

( ⅕ )x  =  ( ⅓ y

3x - 5y = 0 -----(1)

Given : The sum of the two numbers is 16.

y = 16 - x ----(2)

Substitute x = 16 - y into (1)

3x - 5(16 - x) = 0

3x - 80 + 5x = 0

8x - 80 = 0

Substitute x = 10 into (2).

y = 16 - 10

So, the two numbers are 10 and 6.

Problem 4 :

A number between 10 and 100 is five times the sum of its digits. If 9 be added to it the digits are reversed. Find the number. 

Let xy be the number between 10 and 100.

Given : The number between 10 and 100 is five times the sum of its digits.

xy = 5(x + y)

10(x) + 1(y) = 5x + 5y

10x + y = 5x + 5y 

5x - 4y = 0 ----(1)

Given : When 9 is added to the number, the digits are reversed.

xy + 9 = yx  

10(x) + 1(y) + 9 = 10(y) + 1(x)

10x + y + 9 = 10y + x

9x - 9y = -9

x = y - 1 ----(2)

Substitute x = y - 1 into (1).

5(y - 1) - 4y = 0

5y - 5 - 4y = 0

Substitute y = 5 into (2).

Therefore, the required number is 45.

Problem 5 :

One number is greater than thrice the other number by 2. If 4 times the smaller number exceeds the greater by 5, find the numbers.

Let x and y be the two numbers such that x > y .   Given :  One number is greater than thrice the other number by 2.

x = 3y + 2 ----(1)

Given :  4 times the smaller number exceeds the greater by 5.

Substitute x = 3 y + 2.

4y - (3y + 2) = 5

4y - 3y - 2 = 5

Substitute y = 7 into (1).

x = 3(7) + 2

Therefore the two numbers are 23 and 7.

Problem 6 :

A two digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the two digit number.

Let xy be the required two digit number.  Given : The two digit number is 7 times the sum of its digits.

xy = 7(x + y)

10(x) + 1(y) = 7x + 7y

10x + y = 7x + 7y

x = 2y ----(1)

Given : The number formed by reversing the digits is 18 less than the given number.

xy  - yx  = 18 

[10(x) + 1(y)] - [10(y) + 1(x)] = 18

(10x + y) - (10y + x) = 18

10x + y - 10y - x = 18

9x - 9y = 18

Substitute x = 2y into (2).

Substitute y = 2 into (1).

Therefore, the two digit number is 42. 

Problem 7 : 

If a number when divided by 296 gives a remainder 75, find the remainder when 37 divides the same number.

x = 296k + 75

where k is quotient, when x is divided by 296.  In the above sentence we have 296 is multiplied by the constant k , 75 is added to that. In this form , we consider the number 75 as remainder, when the number x is divided by 296.  We want to find the remainder, when we divide the number x by 37. To do this, we need to have 37 at the place where we have 296 in the above equation.  So we can write 296 as 37 times 8 and 75 as 37 times 2 plus 1. It has shown below. 

x = 37×8k + 37×2 + 1 

x = 37(8k + 2) + 1 

Therefore, the remainder is 1 when the number x is divided by 37.

Problem 8 :

Find the least number which when divided by 35, leaves a remainder 25, when divided by 45, leaves a remainder 35 and when divided by 55, leaves a remainder 45.

For each divisor and corresponding remainder, we have to find the difference. 

35 - 25 = 10 

45 - 35 = 10 

55 - 45 = 10 

We get the difference 10 (for all divisors and corresponding remainders)

Find the least common multiple of (35, 45, 55) and subtract the difference from the least common multiple. 

Least common multiple of (35, 45, 55) = 3465 

Therefore, the required least number is

= 3465 - 10

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"Number" Word Problems

What are "number" word problems.

Number word problems involve relationships between different numbers; these exercises ask you to find some number (or numbers) based on those relationships.

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Algebra Word Problems

How do you solve number word problems?

To set up and solve number word problems, it is important clearly to label variables and expressions, using your translation skills to convert the words into algebra. The process of clear labelling will often end up doing nearly all of the work for you.

Number word problems are usually fairly contrived, but they're also fairly standard. Keep in mind that the point of these exercises isn't their relation to "real life", but rather the growth of your ability to extract the mathematics from the English. These exercises are a great way to stretch your mental muscles, use what you know already, apply your logic (and common sense), and then hippity-hop your way to the answer.

What is an example of solving a number word problem?

• The sum of two consecutive integers is 15 . Find the numbers.

They've given me many pieces of information here.

• I'm adding (that is, summing) two things
• the numbers are integers (like −3 and 6 )
• the second number is 1 more than the first
• the result of the addition will be 15

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How do I know that the second number will be larger than the first by 1 ? Because the two integers are "consecutive", which means "one right after the other, not skipping over anything between". (Examples of consecutive integers would be −12 and −11 , 1 and 2 , and 99 and 100 .)

The "integers" are the number zero, the whole numbers, and the negatives of the whole numbers. In going from one integer to the next consecutive integer, I'll have gone up by one unit.

I need to figure out what are the two numbers that I'm adding. The second number is defined in terms of the first number, so I'll pick a variable to stand for this number that I don't yet know:

1st number: n

The second number is one more than the first, so my expression for the second number is:

2nd number: n + 1

I know that I'm supposed to add these two numbers, and that the result will be (in other words, I should set the sum equal to) 15 . This, along with my translation skills, allows me to create an equation, being the algebraic equivalent to "(this number) added to (the next number) is (fifteen)":

n + ( n + 1) = 15

This is a linear equation that I can solve :

2 n + 1 = 15

The exercise did not ask me for the value of the variable n ; it asked for the identity of two numbers. So my answer is not " n  = 7 "; the actual answer, taking into account the second number, too, is:

The numbers are 7 and 8 .

It usually isn't required that you write your answer out like this; sometimes a very minimal " 7, 8 " is regarded as acceptible form. But the exercise asked me, in complete sentences, a question about two numbers; I feel like it's good form to answer that question in the form of a complete sentence.

What do they mean when they say "consecutive even (or odd) integers"?

Some number word problems will refer to "consecutive even (or odd) integers". This means that they're talking about two whole numbers (or their negatives) that are both even or else both odd; in particular, the two numbers are 2 units apart.

• The product of two consecutive negative even integers is 24 . Find the numbers.

I'll start with extracting the information they've given me.

• I'm multiplying (that is, finding the product of) two things
• those two things are numbers
• those two numbers are integers
• those two integers are even
• those two even integers are negative
• the second even integer is 2 units more than the first
• when I multiply, I'll get 24

How do I know that one number will be 2 more than the other? Because these numbers are consecutive even integers; the "consecutive" part means "the one right after the other", and the "even" part means that the numbers are two units apart. (Examples of consecutive even integers are 10 and 12 , −14 and −16 , and 0 and 2 .)

The second number is defined in terms of the first number, so I'll pick a variable for the first number. Then the second number will be two units more than this.

1st number: n 2nd number: n + 2

When I multiply these two numbers, I'm supposed to get 24 . This gives me my equation:

( n )( n + 2) = 24

This is a quadratic equation that I can solve :

( n )( n + 2) = 24 n 2 + 2 n = 24 n 2 + 2 n − 24 = 0 ( n + 6)( n − 4) = 0

This equation clearly has two solutions, being n  = −6 and n  = 4 . Since the numbers I am looking for are negative, I can ignore the " 4 " solution value and instead use the n  = −6 solution.

Then the next number, being larger than the first number by 2 , must be n  + 2 = −4 , and my answer is:

The numbers are −6 and −4 .

In the exercise above, one of the solutions to the exercise — namely, n  = −6 — was one of the solutions to the equation; the other solution to the equation — namely, n  = 4 — had the sign opposite to the other answer to the exercise.

You will encounter this pattern often in solving this type of word problem. However, do not assume that you can use both solutions if you just change the signs to be whatever you think they ought to be. While this often works, it does not always work, and it's sure to annoy your grader. Instead, throw out invalid results, and solve properly for the valid ones.

• Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71 . What are the numbers?

The point of exercises like this is to give me practice in unwrapping and unwinding these words, somehow turning the words into algebraic expressions and equations. The point is in the setting-up and solving, not in the relative "reality" of the exercise. That said, how do I solve this? The best first step is to start labelling.

I need to find two numbers and, this time, they haven't given me any relationship between the two, like "two consecutive even integers". Since neither number is defined by the other, I'll need two letters to stand for the two unknowns. I'll need to remember to label the variables with their definitions.

the larger number:  x

the smaller number:  y

Now I can create expressions and then an equation for the first relationship they give me:

twice the larger:  2 x

three more than five times the smaller:  5 y + 3

relationship between ("is"):  2 x = 5 y + 3

And now for the other relationship they gave me:

four times the larger:  4 x

three times the smaller:  3 y

relationship between ("sum of"):  4 x + 3 y = 71

Now I have two equations in two variables:

2 x = 5 y + 3

4 x + 3 y = 71

I will solve, say, the first equation for x = :

x = (5/2) y + (3/2)

(There's no right or wrong in this choice; it's just what I happened to choose while I was writing up this page.)

Then I'll plug the right-hand side of this into the second equation in place of the x :

4[ (5/2) y + (3/2) ] + 3 y = 71

10 y + 6 + 3 y = 71

13 y + 6 = 71

y = 65/13 = 5

Now that I have the value for y , I can back-solve for x :

x = (5/2)(5) + (3/2)

x = (25/2) + (3/2)

x = 28/2 = 14

As always, I need to remember to answer the question that was actually asked. The solution here is not " x  = 14 ", but is instead the following:

larger number: 14

smaller number: 5

What are the steps for solving "number" word problems?

The steps for solving "number" word problems are these:

• Read the exercise through once; don't try to start solving it before you even know what it says.
• Figure out what you know (for instance, are you adding or multiplying?).
• Figure out what you don't know; this will probably be the value(s) of number(s).
• Pick one or more useful variables for unknown(s) that you need to find.
• Use the variable(s) and the known information to create expressions.
• Use these expressions and the known information to create one or more equations.
• Solve the equation(s) for the unknown(s).
• Check your definition(s) for your variable(s).
• Use this/these definition(s) to state your answer in clear terms.

But more than any list, the trick to doing this type of problem is to label everything very explicitly. Until you become used to doing these, do not attempt to keep track of things in your head. Do as I did in this last example: clearly label every single step; make your meaning clear not only to the grader but to yourself. When you do this, these problems generally work out rather easily.

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Module 10: Linear Equations

Using a problem-solving strategy to solve number problems, learning outcomes.

• Solve number problems
• Solve consecutive integer problems

Solving Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference , of , and and .

The difference of a number and six is thirteen. Find the number.

https://ohm.lumenlearning.com/multiembedq.php?id=142763&theme=oea&iframe_resize_id=mom50

The sum of twice a number and seven is fifteen. Find the number.

Show Solution

https://ohm.lumenlearning.com/multiembedq.php?id=142770&theme=oea&iframe_resize_id=mom60

Watch the following video to see another example of how to solve a number problem.

Solving for Two or More Numbers

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142775&theme=oea&iframe_resize_id=mom70

Watch the following video to see another example of how to find two numbers given the relationship between the two.

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142806&theme=oea&iframe_resize_id=mom80

One number is ten more than twice another. Their sum is one. Find the numbers.

https://ohm.lumenlearning.com/multiembedq.php?id=142811&theme=oea&iframe_resize_id=mom90

Solving for Consecutive Integers

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other.  Some examples of consecutive integers are:

[latex]\begin{array}{c} \hfill \text{…}1, 2, 3, 4, 5, 6\text{,…}\hfill \end{array}[/latex] [latex]\text{…}-10,-9,-8,-7\text{,…}[/latex] [latex]\text{…}150,151,152,153\text{,…}[/latex]

If we are looking for several consecutive numbers, it is important to first identify what they look like with variables before we set up the equation.  Notice that each number is one more than the number preceding it. So if we define the first integer as [latex]n[/latex], the next consecutive integer is [latex]n+1[/latex]. The one after that is one more than [latex]n+1[/latex], so it is [latex]n+1+1[/latex], or [latex]n+2[/latex].

[latex]\begin{array}{cccc}n\hfill & & & \text{1st integer}\hfill \\ n+1\hfill & & & \text{2nd consecutive integer}\hfill \\ n+2\hfill & & & \text{3rd consecutive integer}\hfill \end{array}[/latex]

For example, let’s say I want to know the next consecutive integer after [latex]4[/latex]. In mathematical terms, we would add [latex]1[/latex] to [latex]4[/latex] to get [latex]5[/latex]. We can generalize this idea as follows: the consecutive integer of any number, [latex]x[/latex], is [latex]x+1[/latex]. If we continue this pattern, we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.

We apply the idea of consecutive integers to solving a word problem in the following example.

The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.

The sum of three consecutive integers is [latex]93[/latex]. What are the integers?

• Read and understand:  We are looking for three numbers, and we know they are consecutive integers.
• Constants and Variables: [latex]93[/latex] is a constant. The first integer we will call [latex]x[/latex]. Second integer: [latex]x+1[/latex] Third integer: [latex]x+2[/latex]
• Translate:  The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. “ is 93 ” translates to “[latex]=93[/latex]” since “ is ” is associated with equals.
• Write an equation:  [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]

[latex]x+x+1+x+2=93[/latex]

Combine like terms, simplify, and solve.

[latex]\begin{array}{r}x+x+1+x+2=93\\3x+3 = 93\\\underline{-3\,\,\,\,\,-3}\\3x=90\\\frac{3x}{3}=\frac{90}{3}\\x=30\end{array}[/latex]

• Check and Interpret: Okay, we have found a value for [latex]x[/latex]. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables:

The first integer we will call [latex]x[/latex], [latex]x=30[/latex] Second integer: [latex]x+1[/latex] so [latex]30+1=31[/latex] Third integer: [latex]x+2[/latex] so [latex]30+2=32[/latex] The three consecutive integers whose sum is [latex]93[/latex] are [latex]30\text{, }31\text{, and }32[/latex]

Find three consecutive integers whose sum is [latex]42[/latex].

Watch this video for another example of how to find three consecutive integers given their sum.

• Ex: Linear Equation Application with One Variable - Number Problem. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/juslHscrh8s . License : CC BY: Attribution
• Ex: Write and Solve an Equation for Consecutive Natural Numbers with a Given Sum. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/Bo67B0L9hGs . License : CC BY: Attribution
• Write and Solve a Linear Equations to Solve a Number Problem (1) Mathispower4u . Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/izIIqOztUyI . License : CC BY: Attribution
• Question ID 142763, 142770, 142775, 142806, 142811, 142816, 142817. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Algebra: Number Sequence Word Problems

In these lessons, we will look at solving word problems involving number sequences.

Related Pages Number Sequences Linear Sequences Geometric Sequences Quadratic and Cubic Sequences

Number Sequence Problems are word problems that involve generating and using number sequences. Sometimes you may be asked to obtain the value of a particular term of a sequence or you may be asked to determine the pattern of a sequence.

Number Sequence Problems: Value Of A Particular Term

A number sequence problem may first describe how a sequence of numbers is generated. After a certain number of terms, the sequence will repeat. Follow the description of the sequence and write down numbers in sequence until you can determine how many terms occur before the numbers repeat. Then use that information to determine what a particular term could be.

For example: If we have a sequence of numbers: x, y, z, x, y, z, …. that repeats after the third term, to find the fifth term we find the remainder of 5 divided by 3, which is 2. (5 ÷ 3 is 1 remainder 2).

The fifth term is then the same as the second term, which is y.

Example: The first term in a sequence of numbers is 2. Each even-numbered term is 3 more than the previous term and each odd-numbered term, excluding the first, is –1 times the previous term. What is the 45th term of the sequence?

Solution: Step 1: Write down the terms until you notice a repetition. 2, 5, -5, - 2, 2, 5, -5, -2, …

The sequence repeats after the fourth term. Step 2: To find the 45th term, find the remainder for 45 divided by 4, which is 1. (45 ÷ 4 is 11 remainder 1)

Step 3: The 45th term is the same as the 1st term, which is 2.

Answer: The 45th term is 2.

Number Sequence Problems: Determine The Pattern Of A Sequence

Example: 6, 13, 27, 55, …

In the sequence above, each term after the first is determined by multiplying the preceding term by m and then adding n. What is the value of n?

Solution: Method 1: Notice the pattern: 6 × 2 + 1 = 13 13 × 2 + 1 = 27

The value of n is 1.

Method 2: Write the description of the sequence as two equations with the unknowns m and n, as shown below, and then solve for n.

6m + n = 13             (equation 1)

13m + n = 27           (equation 2)

Using the substitution method Isolate n in equation 1 n = 13 – 6m

Substitute n = 13 – 6m into equation 2 13m + 13 – 6m = 27 7m = 14 m = 2

Substitute m = 2 into equation 1 6(2) + n = 13 n = 1

Answer: n = 1

Solving Number Sequences

This is a method to solve number sequences by looking for patterns, followed by using addition, subtraction, multiplication, or division to complete the sequence. Step 1: Look for a pattern between the given numbers. Step 2: Decide whether to use +, -, × or ÷ Step 3: Use the pattern to solve the sequence.

Examples: 2, 5, 8, 11, _, _, _ 2, 4, 8, 16, _, _, _ 15, 12, 9, _, _, _ 48, 24, 12, _, _, _

Sequences - Find The nth Term

• Describe a linear sequence
• Find the next few terms
• Find the nth term
• Use the nth term to find a term in the sequence

Find the nth term of: a) 6, 11, 16, 21, 26, … b) 2, 10, 18, 26, 34, … c) 8, 6, 4, 2, 0, …

Here are the first five terms of a number sequence. 2, 7, 12, 17, 22 a) (i) Write down the next term in the sequence.     (ii) Explain how you worked out your answer. b) 45 is not a term in this number sequence.     Explain why.

Here are the first five terms of a number sequence. 3, 9, 15, 21 a) (i) Write down the next term in the sequence.     (ii) Explain how you worked out your answer. b) Write down the 7th term in the sequence. c) Jean says 58 is in the sequence.     Is Jean correct?

You must give a reason for your answer.

Find The nth Term Of A Quadratic Sequence

When trying to find the nth term of a quadratic sequence, it will be of the form:

an 2 + bn + c where a, b, c always satisfy the following equations 2a = 2 nd difference (always constant) 3a + b = 2 nd term - 1 st term a + b + c = 1 st term

• Find the n th term T n of this sequence 3, 10, 21, 36, 55, …
• Find the n th term T n of this sequence 0, 7, 20, 39, 64, …

Sequences - Notation

A sequence is a list of numbers that follow a rule.

The nth term of a sequence is given by U n = 3n - 1. Work out: a. The first term. b. The third term. c. The nineteenth term.

The nth term of a sequence is given by U n = n 2 /(n + 1). Work out: a. The first three terms. b. The 49th term.

Sequences - The nth Term - Given A Term Find n

A sequence has nth term given by U n = 5n - 2 Find the value of n for which U n = 153

A sequence has nth term given by U n = n 2 + 5 Find the value of n for which U n = 149

A sequence has nth term given by U n = n 2 - 7n + 12 Find the value of n for which U n = 72

A sequence is generated by the formula U n = an + b where a and b are constants to be found. Given that U 3 = 5 and U 8 = 20 find the values of the constants a and b.

Sequences - Recurrence Relations

Find the first four terms of the following sequence U n + 1 = U n + 4, U 1 = 7

Find the first four terms of the following sequence U n + 1 = U n + 4, U 1 = 5

Find the first four terms of the following sequence U n + 2 = 3U n + 1 - U n , U 1 = 4 and U 2 = 2

A sequence of terms {U n }, n ≥ 1 is defined by the recurrence relation U n + 2 = mU n , where m is a constant. Given also U 1 = 2 and U 2 = 5. a. find an expression in terms of m for U 3 . b. find an expression in terms of m for U 4 . Given the value of U 4 = 21: c. find the possible values of m.

Find The nth Term In A Sequence

A method for finding any term in a sequence

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Base numbers

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Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Babylonian cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) why we count 60 minutes in an hour and 60 seconds in a minute (they might have used it because it has so many multiples, 12 in fact, we wouldn't want any fractions). The Roman system , which didn't have any base system at all, used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000). Imagine how difficult it would be to multiply LXV by MDII! That's why the introduction of the Arabic numeral system , base-10, revolutionized math and science in Europe.

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DU Professor Helps Solve Famous 70-Year-Old Math Problem

Jordyn reiland.

[email protected]

Assistant Professor Mandi Schaeffer Fry is the first faculty member to be published in the Annals of Mathematics since the 1880s.

University of Kaiserslautern Professor Gunter Malle, University of Denver Assistant Professor Mandi Schaeffer Fry and University of Valencia Professor Gabriel Navarro pose for a photo after announcing their theorem in Oberwolfach, Germany.

Whether it be flying trapeze, participating in competitive weightlifting or solving math problems that have confounded academics for decades, Mandi Schaeffer Fry enjoys chasing the next adventure.

Schaeffer Fry, who joined the University of Denver’s Department of Mathematics in the fall of 2023, will be the first faculty member since the 1880s to be published in the Annals of Mathematics , widely seen as the industry’s most prestigious journal.

In 2022, Schaeffer Fry helped complete a problem that dates to 1955—mathematician Richard Brauer’s Height Zero Conjecture.

“Maybe one of the most challenging parts, other than the math itself, was the knowledge of the weight that this would have on the field,” Schaeffer Fry says. “If you’re going to make an announcement like this, you have to be darn sure that it’s absolutely correct.”

Over the years, number crunchers have worked on the problem at universities across the globe, and some found partial solutions; however, the problem was not completed until now.

“Mandi’s accomplishment is exciting. Solving Brauer's Height Zero Conjecture is remarkable,” Mathematics Department Chair Alvaro Arias says. 

The work is also a testament to DU’s achievement as a Research 1 (R1) institution.

Fry and her collaborators—University of Kaiserslautern Professor Gunter Malle, University of Valencia Professor Gabriel Navarro and Rutgers University Professor Pham Huu Tiep—worked around the clock over the course of three months in eight-hour shifts during the summer of 2022 to find a solution.

In April, that work was accepted for publication in the Annals of Mathematics.

'Brauer's Height Zero Conjecture (BHZ) was the first conjecture leading to the part of my field studying 'local-global' problems in the representation theory of finite groups, which seek to relate properties of groups with properties of certain nice smaller subgroups, letting us 'zoom in' on the group using just a specific prime number and simplify things," Schaeffer Fry says. 

"The BHZ gives us a way to tell from the character table of a group (a table of data that encodes lots, but not all, information about the group) whether or not certain of these subgroups, called defect groups, have the commutativity property," she adds.

This paper was especially meaningful to Schaeffer Fry as she had always wanted to work with Malle, Tiep and Navarro as they have been her primary mentors. Tiep was her PhD advisor and this was the first time they had worked together since then.

Fry believes she has solidified her place in the field and knows she’ll likely never top this accomplishment, but she’s always looking for the next adventure—whether that’s in or out of the classroom.

Flying high and pumping iron

When Schaeffer Fry isn’t on DU’s campus working with students or conducting research, you can find her flying trapeze and competitive weightlifting.

Schaeffer Fry became involved in competitive weightlifting during graduate school, and, in the last year of her PhD at the University of Arizona, she defended her dissertation one day and got on a plane and competed at the national level for “university-aged” athletes—which included Olympians.

While she now lifts weights more casually, Schaeffer Fry competed last September in an over-35 competition and qualified for the USA Weightlifting Masters National Championships.

It was a “field trip” during a conference in Berkeley, California, in 2018 that led Fry to become enamored with flying trapeze.

In fact, she enjoyed it so much she signed up to be a member of Imperial Flyers, an amateur flying trapeze cooperative located in Westminster. Once she found out about the sport, her previous experience as a gymnast made it a natural fit.

Not only is she working on her own intermediate tricks, she’s also a “teaching assistant” at Fly Mile High, the state’s only flying trapeze and aerial fitness school.

“It’s exhilarating; it’s gotten me a bit over my fear of heights,” she says.

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7 big issues at stake in the 2024 election

WASHINGTON — The policy contrasts between President Joe Biden and former President Donald Trump are sharpening as the general election campaign gets fully underway.

But what does the choice represent for ordinary voters and the economic and cultural issues they care about? A rematch between the Democratic incumbent and his Republican predecessor may feel uninspiring to many voters, but the policy stakes are enormous for tens of millions of Americans — and the world.

Here are seven big issues at stake in the 2024 election.

The contrast: Biden favors federal abortion protections; Trump opposes them. Trump supported nationwide restrictions on abortion as president but now downplays the need for a federal ban, as Republicans are divided over the issue. Biden does not support federal limits.

Biden has championed the Women’s Health Protection Act, a bill to protect abortion rights in all 50 states under federal law and prohibit medically unnecessary hurdles to accessing the procedure. He has asked voters to send him a Democratic Congress that supports legal abortion to achieve that.

Trump has boasted that he "broke Roe v. Wade" by picking three of the five Supreme Court justices who overturned it, delivering on a four-decade goal of the GOP. More recently, Trump has openly fretted that the backlash may cost him and his party the election. Last week, Trump said the issue should be left to states, a shift from his support for nationwide restrictions when he was president. His new stance has drawn pushback from GOP allies, like Sen. Lindsey Graham, of South Carolina , and anti-abortion-rights advocates, who say that he is wrong and that Republicans should not be deterred from their long-standing goal of enacting some nationwide abortion limits.

Some Republicans downplay the prospects of federal abortion restrictions’ passing Congress, even if they win full control. Biden and his allies are telling voters to look at the GOP’s long history of championing federal restrictions and not their recent rhetoric.

Immigration

The contrast: Trump has promised a sweeping crackdown on illegal immigration and tougher executive actions; Biden is asking Congress to give him more tools to manage an overwhelmed border and create new legal pathways to immigrate to the U.S.

Trump has called existing border laws an existential threat to the U.S., saying migrants are “ poisoning the blood of our country” and bringing new “ languages .” His campaign website says: “President Trump will shut down Biden’s border disaster. He will again end catch-and-release, restore Remain in Mexico , and eliminate asylum fraud. In cooperative states, President Trump will deputize the National Guard and local law enforcement to assist with rapidly removing illegal alien gang members and criminals.”

After having rescinded some of Trump's policies, Biden has recently shifted to support stricter immigration laws as the system remains overwhelmed. He championed a bipartisan bill to raise the bar for gaining asylum, grant more U.S. resources to process asylum claims and turn away migrants who do not qualify, and empower the president to temporarily shut down the border if migration levels hit certain triggers. (Republicans blocked the bill in the Senate amid lobbying by Trump , who wants to use the border as an election issue.) Biden has also endorsed the U.S. Citizenship Act , which would grant a pathway to citizenship for people in the U.S. illegally if they pass background checks and pay their taxes.

Fundamentally, Trump has aligned with forces who want less immigration into the country, while Biden has embraced the belief that immigrants make the U.S. better.

Health care and prescription drugs

The contrast: Biden wants to extend Affordable Care Act provisions and empower Medicare to negotiate more prescription drugs; Trump has aggressively criticized the ACA but not offered a health care plan.

Biden, who was vice president when the Affordable Care Act passed in 2010, sees it as a cherished achievement to protect and strengthen. The law, also known as "Obamacare," which has extended coverage to 45 million people through subsidies, insurance mandates and a Medicaid expansion, continues to face conservative opposition.

Separately, Biden has touted a provision in his party-line Inflation Reduction Act that empowers Medicare to negotiate lower prices for 10 prescription drugs. He said he wants to boost that to 50 if he is re-elected, with the goal of $200 billion in savings.

Trump spent his four years as president fighting unsuccessfully to repeal and unravel the law — through legislation and executive action and endorsing lawsuits to wipe it out. In November, Trump called for revisiting plans to "terminate" the ACA . He has recently sought to downplay that and insists he only wants to improve the law. But he has not offered a health care plan. Many of his GOP allies in Congress still favor repealing or undoing the ACA, including a budget by the Republican Study Committee, which boasts about 80% of the House GOP conference as members, including Speaker Mike Johnson, of Louisiana.

The contrast: Trump's 2017 tax cuts expire at the end of next year, and he has called for extending them; Biden has called for raising taxes on families earning over $400,000 to fund various priorities.

A series of Trump tax cuts, which Republicans passed on a party-line basis in 2017, expire at the end of 2025. Congress and the winner of the election will decide what happens to them.

In a recent private speech to wealthy donors, Trump s aid his policies include "extending the Trump tax cuts" if he is elected, according to a Trump campaign official. That would preserve lower rates across the income spectrum, with the biggest benefits for top earners.

Biden has attacked that law as a giveaway to the wealthiest Americans, vowing to make "big corporations and the very wealthy finally pay their fair share." He has backed a corporate tax rate hike from 21% to 28% and said that "nobody earning less than $400,000 will pay an additional penny in federal taxes." Biden is also calling for a $3,600-per-child tax cut for families, an $800 average tax break for "front-line workers" and a 25% minimum tax on billionaires, according to a newly released campaign plank.

The expiration of the Trump tax cuts will restore the unlimited federal deduction for state and local taxes, which Republicans had capped at $10,000 in the 2017 law. Republicans broadly support preserving the cap, with some exceptions, while most Democrats want to lift it.

Judges and the Supreme Court

The contrast: Their track records tell a clear story. Trump has picked young conservative judges to serve on the federal bench, while Biden has picked liberals with a focus on professional and personal diversity.

One of the clearest contrasts is what kinds of judges Trump and Biden would pick for lifetime appointments on the federal courts. A simple way for voters to think about it is whether they prefer new judges with the conservative views of Justice Neil Gorsuch, Trump's first Supreme Court pick, or with the liberal views of Justice Ketanji Brown Jackson, Biden's (so far only) high court pick.

As president, Trump nominated young conservative judges who will serve for generations. Biden has focused on finding judges with diverse backgrounds and résumés, including more civil rights lawyers and public defenders.

Perhaps the biggest question is whether a Supreme Court vacancy will open up in the next four years. The presidential election winner and the party that controls the Senate would fill it.

The contrast: Trump is pushing a 10% across-the-board tariff on imports; Biden's White House opposes that, saying it would raise inflation.

Trump, long a skeptic of U.S. trade deals, has proposed to impose a 10% tariff on all imported goods if he returns to the White House. He recently told Fox News that it could be 60% — or potentially “more than that” — on imports of Chinese goods.

Biden opposes that idea. In a memo over the weekend, the White House slammed the idea of "across-the-board tariffs that would raise taxes and prices by $1,500 per American family," without naming Trump; it referred to an estimate by the Center for American Progress, a liberal think tank, that Trump's 10% tax on imports could cost an average American household $1,500 per year.

Biden, instead, has sought to boost domestic manufacturing with major federal investments in semiconductors and electric vehicles.

Foreign policy and NATO

The contrast: Biden favors Ukraine aid, while Trump is skeptical of it; Biden supports NATO and a traditional view of American power, while Trump has criticized NATO and voiced some isolationist views.

The clearest example of the foreign policy differences between the two concerns the fate of Ukraine, which is running low on ammunition and says it needs U.S. assistance to continue holding off Russia’s aggression. Biden is an ardent proponent of helping Ukraine, while Trump has poured cold water on U.S. aid to Ukraine and successfully pressured House Republicans to block it since they took the majority in January 2023.

And that points to a deeper divide: Biden is an outspoken supporter of the NATO alliance as a bulwark against adversaries like Russia and China and of preserving the post-World War II order. Trump has dialed up his criticisms of NATO and aligned with a growing isolationist wing in the U.S. that wants to be less involved in global affairs. Trump recently said that as president, he “would encourage” Russia “to do whatever the hell they want” to member countries who are “delinquent” in their dues.

Sahil Kapur is a senior national political reporter for NBC News.

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Mathematics > Combinatorics

Title: on the min-max star partitioning number.

Abstract: In this paper, we introduce a novel star partitioning problem for simple connected graphs $G=(V,E)$. The goal is to find a partition of the edges into stars that minimizes the maximum number of stars a node is contained in while simultaneously satisfying node-specific capacities. We design and analyze an efficient polynomial time algorithm with a runtime of $\mathcal{O}(|E|^2)$ that determines an optimal partition. Moreover, we explicitly provide a closed form of an optimal value for some graph classes. We generalize our algorithm to find even an optimal star partition of linear hypergraphs, multigraphs, and graphs with self-loop. We use flow techniques to design an algorithm for the star partitioning problem with an improved runtime of $\mathcal{O}(\log(\Delta) \cdot |E| \cdot \min\{|V|^{\frac{2}{3}},|E|^{\frac{1}{2}}\})$, where $\Delta$ is maximum node degree in $G$. In contrast to the unweighted setting, we show that a node-weighted decision variant of this problem is \texttt{strongly NP-complete} even without capacity constraints. Furthermore, we provide an extensive comparison to the problem of minimizing the minimum indegree satisfying node capacity constraints.

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