Lesson 3 · from §36 of the 1910 text
Problems with Two or More Unknowns
These problems have two or more unknowns. You can always represent one by a single letter and the rest by binomials made from that letter and some numbers. Wrap each binomial in parentheses and treat it like a single number. Once you handle a binomial as easily as a single letter like x, you've made real progress in thinking algebraically.
Example. The sum of two numbers is 88. Three times the less equals twice the greater, plus 29.
Let n be the less; then 88 - n is the greater. 3n = 2(88 - n) + 29 \;\Rightarrow\; 3n = 205 - 2n \;\Rightarrow\; 5n = 205 \;\Rightarrow\; n = 41. So the numbers are 41 and 47. Check: 41 + 47 = 88, and 3 \cdot 41 = 2 \cdot 47 + 29 = 123. ✓
Coins. 15 nickels and dimes are worth 90 cents. Let d be the dimes; then 15 - d are nickels, and 10d + 5(15 - d) = 90, so d = 3 dimes and 12 nickels.
Interest. Part of $800 is at 3% and the rest at 4%, earning $30 a year. Let x be the part at 3%: 0.03x + 0.04(800 - x) = 30. Multiply by 100 to clear decimals: 3x + 4(800 - x) = 3000, so x = 200 and the other part is 600.
Quick check
Twenty coins, all nickels and dimes, are worth $1.40. How many of each?
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 every unknown with one letter, write an equation, solve, and check.
The sum of two numbers is 88. Three times the less equals twice the greater, plus 29. Find the numbers.
The sum of two numbers is 49. Twice the greater, minus 13, equals five times the less. Find the numbers.
The sum of two numbers is 143. Ten times the less added to five times the greater equals 950. Find the numbers.
Separate 45 into two parts so that five times the greater plus four times the less equals 207.
The sum of two numbers is 88. Three times the greater equals five times the less, plus 29. Find the numbers.
Separate 93 into two parts so that seven times the less, minus 7, equals six times the greater.
Separate 48 into two parts so that twice the greater, minus 7, equals three times the less, minus 5.
The sum of two numbers is 12\tfrac12. Seven times one number minus ten times the other equals 45. Find the numbers.
Separate 121 into two parts so that four times one, increased by 8, equals three times the other.
Twice a certain number minus five times another number equals 240. The sum of the numbers is 15. Find the numbers.
The sum of two numbers is 14. Nine times one minus eleven times the other equals zero. Find the numbers.
The square of a number plus the square of the next consecutive number is 17 greater than twice the square of the smaller number. Find the numbers.
The difference of the squares of two consecutive numbers is 75. Find the numbers.
The difference of the squares of two consecutive numbers is 23. Find the numbers.
The difference of the squares of two consecutive odd numbers is 104. Find the numbers.
The difference of the squares of two consecutive odd numbers is 40. Find the numbers.
The product of two consecutive even numbers is 56 less than the square of the greater number. Find the numbers.
The product of two consecutive odd numbers equals the square of the smaller increased by 46. Find the numbers.
A square has the same area as a rectangle whose length is 8 inches greater and whose breadth is 4 inches less than the side of the square. Find the side of the square and the area of each.
A square field has the same area as a rectangular field whose length is 30 rods greater and whose breadth is 20 rods less than the side of the square. How many acres are in each field? (One acre is 160 square rods.)
A tennis court for two players is 24 feet longer than twice its breadth. The distance around the court is 210 feet. Find the length and breadth.
A tennis court for four players is 6 feet longer than twice its breadth. Its perimeter is 228 feet. Find the dimensions.
The breadth of a basketball court is 20 feet less than its length. The perimeter is 80 yards. Find the dimensions in feet.
The perimeter of a football field is 780 feet. Its length is 50 yards less than three times its breadth. Find the length and breadth in feet.
Fifteen coins, all nickels and dimes, are worth 90 cents. Find the number of each.
Thirty-eight coins, all dimes and quarters, are worth $5.30. Find the number of each.
A collection of nickels, dimes, and quarters is worth $6.05. There are 5 more nickels than dimes, and the number of quarters equals the nickels and dimes together. Find the number of each.
Forty coins, all nickels and dimes, are worth $2.90. Find the number of each.
A is 20 years older than B. In 10 years A will be twice as old as B. Find each age now.
A is four times as old as B. In 20 years A will be twice as old as B. Find each age now.
A's age is 8 years more than twice B's. Sixteen years ago A was four times as old as B. Find each age now.
Part of $800 is invested at 3% and the rest at 4%. The yearly income from both is $30. Find each investment.
Part of $1400 is invested at 5% and the rest at 6%. The total yearly income is $76. Find each investment.
A sum at 6% and a second sum at 8% yield $53 a year in all. The first sum exceeds the second by $125. Find each.
A 5% investment yields $15 a year less than a 6% investment. Together the two investments are $1240. Find each.