Lesson 3 · from §23 of the 1910 text
Solving Problems
Solving a word problem takes five steps:
- Read the problem carefully.
- Represent the unknown number by a letter.
- Express the conditions in the problem as an equation with that letter.
- Solve the equation.
- Check by putting your value back into the problem (the words), not just the equation. Checking the equation can't catch a mistake you made translating the words.
Sometimes you can translate almost word for word:
\underbrace{4n}_{\text{four times a number}} \; \underbrace{-\,6}_{\text{diminished by 6}} \; \underbrace{=}_{\text{gives the same as}} \; \underbrace{n + 30}_{\text{the number increased by 30}}
Seven times A's age two years ago equals five times his age ten years from now becomes 7(x - 2) = 5(x + 10).
Two unknowns, one letter. The sum of two numbers is 67 and their difference is 5. Let n be the smaller; then the greater is n + 5. So n + n + 5 = 67, 2n = 62, n = 31, and the greater is 36. Check: 31 + 36 = 67 and 36 - 31 = 5. ✓
Beyond these directions, the best help is studying worked examples, reading each problem carefully once or twice, and using common sense.
Quick check
The sum of two numbers is 50 and their difference is 14. Find them.
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.
Solve and check. Use the five steps.
To what number must 22 be added so that the sum may be 50?
From what number must 15 be subtracted so that the remainder may be 47?
What number increased by 9 equals 28?
What number diminished by 17 equals 35?
What number, doubled and then diminished by 27, gives 49?
What number, tripled and then diminished by 36, gives twice the original number?
Three times a certain number, less 17, equals twice the number, less 1. Find the number.
Five times a certain number, increased by 6, equals twice the number, increased by 15. Find the number.
Four times a certain number, plus 9, equals seven times the number, minus 33. Find the number.
A certain number added to 9 gives the same result as the number subtracted from 71. Find the number.
If 6 is added to twice a number, and 10 is subtracted from four times the number, the results are the same. Find the number.
The sum of two numbers is 67, and their difference is 5. Find the numbers.
The sum of two numbers is 74, and their difference is 12. Find the numbers.
The sum of two numbers is 45; the second is 3 less than the first. Find the numbers.
The sum of two numbers is 44, and one exceeds the other by 8. Find the numbers.
The sum of three numbers is 83. The second is 4 less than the first, and the third is 9 greater than the first. Find the numbers.
The sum of three numbers is 66. The second is 3 less than the first, and the third is 18 greater than the second. Find the numbers.
The sum of two consecutive numbers is 37. Find the numbers.
Find three consecutive numbers whose sum is 39.
Find four consecutive numbers whose sum is 90.
Find two consecutive even numbers whose sum is 30.
Find three consecutive odd numbers whose sum is 87.
Find four consecutive even numbers whose sum is 100.
A rectangle whose perimeter is 38 feet is 3 feet longer than it is wide. Find its dimensions.
A rectangle whose perimeter is 128 feet is 16 feet longer than it is wide. Find the dimensions.
The length of a rectangle is 7 feet more than twice the width. Its perimeter is 104 feet. Find the dimensions.
The length of a rectangle is 5 feet more than four times its width. Its perimeter is 90 feet. Find the dimensions.
At Pittsburgh on June 21 the day is 6 hours 6 minutes longer than the night. How long is the night? the day? (Answer in hours; 6 minutes is 0.1 hour.)
A's age is twice B's, and C is 7 years older than A. The sum of their ages is 67 years. Find the age of each.
A's age is three times B's, and C is 10 years older than B. Five years from now the sum of their ages will be 60 years. Find the age of each now.
A is 10 years older than B, and C is 6 years younger than B. Four years ago the sum of their ages was 46 years. Find the age of each now.
A's age is 2 years more than twice B's, and C's age is 7 years less than A's. Six years from now the sum of their ages will be 70 years. Find the age of each now.
In 1907 the corn crop of the United States exceeded the oats crop by 1838 million bushels, and the wheat crop was 120 million bushels less than the oats crop. The total was 3981 million bushels. Find each crop in millions of bushels.
The north and south frigid zones have the same width, as do the north and south temperate zones. The torrid zone is 47 degrees wide, twice the width of the north frigid zone. All five zones together are 180 degrees wide. Find the width of each kind of zone.
Taking all the people in the United States as 100%, 12% more were engaged in agriculture than in industry, 8% more in industry than in commerce, and the rest, 24%, in other pursuits. Find the percent in agriculture, industry, and commerce.
One year Montana produced 110 million pounds of copper more than Michigan, and 139 million pounds more than Arizona. Together the three produced 514 million pounds. Find each state's production in millions of pounds.
The Eiffel Tower is 120 feet less than twice the height of the Washington Monument. The Monument is 105 feet higher than the Great Pyramid and 107 feet higher than St. Peter's in Rome. Their heights add to 2443 feet. Find the height of each.
The coal fields of China and Japan are 6000 square miles larger than those of the United States. The U.S. fields exceed twice the area of all other countries' (besides China and Japan) by 38,400 square miles. The world total is 471,800 square miles. Find each area.
One year the U.S. produced 5573 tons of copper less than five times the output of Spain and Portugal. Those two produced 808 tons less than twice Japan's output. All other countries produced 4615 tons less than five times Japan's output. The world produced 486,084 tons. Find each output.
Asia is 982,000 square miles more than twice the area of North America. North America exceeds South America by 1,186,000 square miles and Europe by 4,383,000 square miles. The four continents total 35,692,000 square miles. Find the area of each.
U.S. troops in the Civil War numbered 15,621 less than nine times those in the Revolution, which was 266,841 less than in the War of 1812. The three wars together involved 3,658,811 troops. Find the number in each.
St. Peter's Cathedral (Rome) holds 29,000 more people than St. Paul's Cathedral (London), and 17,000 and 22,000 more than the Cathedral at Milan and St. Paul's church (Rome). Together they hold 148,000. Find each capacity.