Lesson 3 · from §19 of the 1910 text
Word Problems
Now put translation and solving together. Here's the method on a sample:
One number is three times another, and their sum is 64. Find the numbers.
- Choose a letter for one unknown: let x be the smaller number.
- Represent the others with it: the larger is 3x.
- Write an equation from the facts: x + 3x = 64.
- Solve it: 4x = 64, so x = 16, and the larger number is 3x = 48.
- Check against the original words: 48 is three times 16, and 16 + 48 = 64. ✓
Always check against the words of the problem, not just your equation. If you wrote the equation wrong, it can still "check."
Quick check
One number is 7 greater than another, and their sum is 39. Find both numbers.
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.
Choose a letter for one unknown, write the others in terms of it, form an equation, and solve.
One number is twice another, and the sum of the two is 135. Find both numbers.
One number is four times another, and the sum of the two is 105. Find both numbers.
One number is five times another, and the difference of the two is 52. Find both numbers.
One number is 5 greater than another, and their sum is 129. Find both numbers.
One number is 18 greater than another, and their sum is 168. Find both numbers.
A rectangle is five times as long as it is wide, and its perimeter is 156 feet. Find the length and the width.
A rectangle is 12 feet longer than it is wide, and its perimeter is 96 feet. Find the length and the width.
The perimeter of a rectangle is 98 feet, and its length is 4 feet more than twice the width. Find the length and the width.
Find the dimensions of a rectangle whose perimeter is 88 feet and whose length is 20 feet less than three times the width.
Find three consecutive numbers whose sum is 45.
Find four consecutive numbers whose sum is 106.
Find five consecutive numbers whose sum is 85.
Find three consecutive odd numbers whose sum is 291.
Find three consecutive even numbers whose sum is 66.
Find five consecutive odd numbers whose sum is 315.
A's age is two years less than B's, and the sum of their ages is 43 years. Find the age of each.
B's age is twice A's. Six years from now the sum of their ages will be 54 years. How old is each now?
In 1910 the United States had 52,000 more miles of railway than Europe, and together they had 402,000 miles. Find the mileage of each.
The Nile is 500 miles longer than the Amazon and 300 miles shorter than the Mississippi (entire). The three lengths add to 11,500 miles. Find the length of each, using the book's figures.
Mount McKinley is 6313 feet higher than Pike's Peak. The sum of their heights is 34,607 feet, which is 5605 feet more than the height of Mount Everest. Find the height of each, using the book's figures.
The combined horsepower of a Mallet Compound freight engine (Erie R.R.), a Pacific passenger engine (Pennsylvania R.R.), and a Baltimore and Ohio electric tractor is 11,200. The freight engine has 1800 more horsepower than the electric tractor and 1000 less than the passenger engine. Find the horsepower of each.