Lesson 4 · from §86 of the 1910 text
Problems in Two Unknowns
Problems with two unknowns
In these problems, state two equations in two unknowns. Instead of x and y, use the first letter of the word for each unknown: in Problem 6 below, n is the number of nickels and q the number of quarters. This is worth doing in every problem with two or more unknowns, unless two of the words begin with the same letter.
Digit problems
Example. The sum of the two digits of a 2-digit number is 9. If 45 is subtracted from the number, the result is expressed by the digits in reverse order. Find the number.
Let t = the digit in tens' place and u = the digit in units' place. Then t + u = 9. \qquad (1) The digit t, standing in tens' place, is worth 10t. So the number is 10t + u, and the number with its digits reversed is 10u + t. Hence 10t + u - 45 = 10u + t, \quad\text{or}\quad t - u = 5. \qquad (2) Solving (1) and (2), t = 7 and u = 2: the number is 72. Check: 7 + 2 = 9 and 72 - 45 = 27. ✓
Levers and money
Two weights balance when (weight) × (distance from the fulcrum) is the same on both sides. Money at interest earns (principal) × (rate) a year: 400 at 6% earns0.06 \times 400 = 24$ dollars.
Try one:
Quick check
The sum of two numbers is 30 and their difference is 8. Find the 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.
State two equations in two unknowns, using the first letter of each unknown's name, and solve.
The difference of two numbers is 25 and their sum is 46. Find the numbers.
The quotient of two numbers is 6 and their sum is 49. Find the numbers.
Find two numbers whose difference is 36 and whose quotient is 3.
The value of a certain fraction is \frac35. If 3 is added to the numerator and 1 to the denominator, the value of the resulting fraction is \frac34. Find the fraction.
The greater of two numbers divided by the less gives a partial quotient of 3 and a remainder of 6. The less divided by the greater gives a fraction which is .7 less than 1. Find the numbers.
A collection of nickels and quarters, containing 77 coins, amounted to $9.85. How many coins of each kind were there?
If \frac43 is subtracted from the numerator and \frac19 added to the denominator of a certain fraction, the value of the resulting fraction is \frac35. The sum of the numerator and the denominator of the original fraction is 19. Find the fraction.
The difference between the numerator and the denominator of a certain proper fraction is 24. If \frac38 is added to the numerator and \frac14 taken from the denominator, the value of the resulting fraction is \frac{3}{14}. Find the fraction.
Two weights balance when one is 12 inches and the other 8 inches from the fulcrum. If the first weight increased by 2 pounds is placed 10 inches from the fulcrum, the balance is maintained. Find the two weights.
Two weights balance when one is 12 inches and the other 18 inches from the fulcrum. If the first weight is decreased 12 pounds, the other weight must be moved 3 inches nearer the fulcrum to balance. Find the weights.
Two weights balance when one is 15 inches and the other 10 inches from the fulcrum. The smaller weight is moved one inch nearer the fulcrum and decreased 6 pounds. Then the larger weight is decreased 12 pounds and a balance results. Find the two weights.
A's age is now twice B's. Seven years ago B was \frac13 as old as A. Find the age of each now.
In 5 years A will be twice as old as B. Five years ago A was three times as old as B. Find the age of each now.
The perimeter of a rectangle is 232 feet and the length is 8 feet more than twice the width. Find the dimensions of the rectangle.
A part of $1000 is invested at 6% and the remainder at 5%. The yearly income from both is $54. Find the number of dollars in each investment.
A part of $2000 is invested at 4\frac12% and the remainder at 3\frac12%. The yearly income from the 3\frac12% investment exceeds the other yearly income by $10. Find the number of dollars in each investment.
A part of $5000 is invested at 4% and the remainder at 6%. The 4% investment yields $126 more in 5 years than the one at 6% does in 3 years. Find the number of dollars in each investment.
Ten rubles are worth 10 cents less than 20 marks, and 12 marks are worth 4 rubles and a dollar. Find the value of a ruble and a mark in cents.
Five francs are worth 19 cents more than 2 florins, and the sum of 3 francs and one florin is worth 5 cents less than one dollar. Find the value of a florin and a franc in cents.
Digit problems (Problem 20 is the worked example above).
The sum of the digits of a 2-digit number is 7. If 27 is added to the number, the result is expressed by the digits in reverse order. Find the number.
The tens' digit of a 2-digit number is twice the units' digit. If 36 is subtracted from the number, the result is expressed by the digits in reverse order. Find the number.
If a 2-digit number is divided by the sum of its digits, the quotient is 4. If 36 is added to the number, the result is expressed by the digits in reverse order. Find the number.
If a 2-digit number is increased by 3 and then the result is divided by the sum of its digits, the quotient is 9. If the number is divided by three times the units' digit, the quotient is 17. Find the number.
If a 2-digit number is divided by the sum of its digits, the quotient is 7. If the number formed by the digits in reverse order is divided by the sum of the digits and 3, the quotient is 3. Find the number.