Lesson 2 · from §22 of the 1910 text
Representing Numbers
Before tackling bigger word problems, practice writing quantities with letters.
- Greater than n by 5 is n + 5; less than n by 3 is n - 3.
- If one part of 10 is 6, the other part is 10 - 6. If one part of x is a, the other part is x - a.
- If two numbers add to 18 and one is n, the other is 18 - n.
- The excess of 10 over x is 10 - x: how much bigger the first is.
- If someone is n years old, in 4 years they'll be n + 4, and 3 years ago they were n - 3.
Quick check
The sum of two numbers is 50. If one of them is k, what is the other?
Reading equations aloud
Translation works in both directions. Try reading these out loud in plain English, calling m "a number" and n "a second number":
n - 2 = 8 (a number less 2 is 8), \;3n = 27, \;18 - n = n - 4, \;m + n = 20, \;2m = n + 6, \;3m - 2n = 8, \;m - b = n, \;80 - m = 30 + n.
Quick check
Write as an equation: twice a number, m, is 6 more than a second number, n.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Represent each number.
Represent a number: Greater than n by 5.
Represent a number: Greater than n by a.
Represent a number: Less than n by 3.
Represent a number: Less than n by b.
Represent a number: Four times n.
Represent a number: Three greater than a + b.
Represent a number: c greater than a + b.
Represent a number: Five less than 2a - b.
Represent a number: c less than 2a - b.
Represent a number: b less than two times n.
Represent a number: a greater than five times n.
Represent a number: Seven less than four times n.
Represent a number: Eight greater than three times n.
One part of 10 is 6. What is the other part?
One part of x is 4. What is the other part?
One part of 12 is y. What is the other part?
One part of x is a. What is the other part?
One part of a is x. What is the other part?
One part of x + y is z. What is the other part?
The sum of two numbers is 18. If one of them is 7, what is the other?
The sum of two numbers is 18. If one of them is n, what is the other?
The difference of two numbers is 18. If the greater is 34, what is the other?
The difference of two numbers is 18. If the greater is n, what is the other?
The sum of two numbers is 30. If the smaller is b, what is the other?
The sum of two numbers is a. If one of them is 7, what is the other?
The sum of two numbers is a. If one of them is x, what is the other?
The difference of two numbers is d. If one of them is b, what is the other? (Either answer can be right, depending on which number is greater.)
The difference of two numbers is d. If one of them is n, what is the other? (Either answer can be right.)
What is the excess of 10 over 4? of 10 over x?
By how much does 25 exceed 9? 25 exceed y? 16 exceed a + b?
How much greater is 40 than 27? than 40? than a? How much greater is a than b?
How much smaller is 22 than 36? 14 than a? x than y?
By how much does a + b exceed a - b? a + 6 exceed b - 6? 4x - 3 exceed 3x - 4?
A is n years old. What will his age be 4 years from now? x years from now? What was it 3 years ago?
A's age is 2n - 3 years. What will it be 10 years from now? a years from now? What was it 8 years ago? a years ago?
A and B each have x dollars. If A gives B four dollars, how much will each have?
A and B each have x + 50 dollars. If B gives A y dollars, how much will each have?
A has x + 30 dollars and B has 3x - 4 dollars. Express as equations: (a) Together they have $200. (b) A has as many dollars as B. (c) A has $10 less than B. (d) If A gains $100 and B loses $50, they have equal amounts.
A's age is x years, B's is 2x + 7, and C's is 3x - 8. Express (a) each age five years from now and (b) each age three years ago. Then write equations for: (c) In four years, A's and B's ages will add to 40. (d) Six years ago, C was 24 years older than A. (e) In 10 years A will be as old as B is now. (f) Four years ago C was as old as B will be in 10 years. (g) In x years B will be 40. (h) In 2 years their three ages will add to 100.