Lesson 10 · from §88 of the 1910 text
Problems in Three and Four Unknowns
Problems with three unknowns
State three equations in three unknowns (four in four), then solve.
Facts from geometry. The three angles of any triangle add up to 180 degrees. The four angles of any quadrilateral (a closed figure bounded by four straight lines) add up to 360 degrees.
Example. Find three numbers such that the first and second add to 20, the second and third to 26, and the first and third to 24. With x, y, z: x + y = 20, y + z = 26, x + z = 24. Adding all three, 2(x + y + z) = 70, so x + y + z = 35. Subtracting each equation from this: z = 15, x = 9, y = 11.
For a 3-digit number with digits h, t, u, the number is 100h + 10t + u.
Try one:
Quick check
The second angle of a triangle is twice the first, and the third is three times the first. Find the angles.
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 three equations in three unknowns (four in four) and solve.
Find three numbers of which the sum of the first and second is 76, the second and third 54, and the first and third 58.
The sum of three numbers is 58. The sum and the quotient of two of them are 24 and 2 respectively. Find the numbers.
The perimeter of a triangle is 64 feet. Two of its sides are equal, and the third side is 10 feet longer than either of the first two. Find the length of each side.
The sum of two sides of a triangle is 52 feet and the difference is 12 feet. The perimeter of the triangle is 93 feet. Find the length of each side.
The sum of the two sides of a triangle which meet at one vertex is 41 feet, at another vertex 48 feet, and at the third vertex 43 feet. Find the length of each side.
The sum of three numbers is 26. The quotient of two of them is 9, and the sum of these two divided by the third is 3\frac13. Find the numbers.
The angles of a triangle add up to 180 degrees.
Two of the angles of a triangle are equal, and their sum is equal to the third. Find the number of degrees in each angle.
Two angles of a triangle are equal, and their sum is \frac12 the third. How many degrees are there in each?
Angle A of a triangle is 17 degrees larger than angle B, and angle B is 20 degrees larger than angle C. How many degrees are there in each?
The sum of two angles of a triangle is 36 degrees more than the third, and the third is six times the difference of the first two. How many degrees are there in each?
Work problems
A and B together can do a piece of work in 2 days, A and C in 3 days, and B and C in 4 days. Find the time required by each alone and by all together.
Two pumps together can fill a tank in 4 hours. The first of these and a third together can fill the tank in 6 hours. All three together can fill the tank in 3\frac{9}{13} hours. Find the number of hours required by each alone.
Fractions and digits
The sum of two fractions having the same denominator is 6. If 1 is added to the numerator of the first, and 1 is subtracted from the numerator of the second, the resulting fractions will be equal. If 22 is added to the denominator of each fraction, the sum of the resulting fractions is \frac12. Find the fractions.
The sum of the digits of a 3-digit number is 15. The units' digit exceeds the tens' digit by 5. If 396 is added to the number, the result is expressed by the digits in reverse order. Find the number.
If the tens' and units' digits of a 3-digit number are interchanged, the resulting number is 54 greater than the number. If the tens' and hundreds' digits are interchanged, the resulting number is 360 less than the number. The sum of the digits is 10. Find the number.
The sum of the 4 digits of a 4-digit number is 9. The units' digit is twice the thousands' digit, and the tens' digit equals the hundreds' digit. If 2997 is added to the number, the result is expressed by the digits in reverse order. Find the number.
The angles of a quadrilateral add up to 360 degrees.
Find the number of degrees in each angle of a quadrilateral in which the sum of the first and second angles is 200 degrees, the sum of the second and third 180 degrees, and the sum of the second, third, and fourth 255 degrees.
The sum of two opposite angles of a quadrilateral is 180 degrees and their difference is 30 degrees. The difference of the other two angles is 36 degrees. Find each angle.
The sum of two opposite sides of a quadrilateral is 30, the sum of the other two sides is 35, and two adjacent sides are equal. The sum of the equal sides is less by 17 than the sum of the other two. Find each side.
Money
A, B, and C had together $300. A gave to B and C as many dollars as each of them had, after which B gave to A and C as many dollars as each of them then had. They then had equal amounts. How many dollars had each at first?
A, B, and C had together $192. A gave to B and C as many dollars as each of them had, after which B gave to A and C as many dollars as each of them then had; and, lastly, C gave to A and B as many dollars as each of them then had. They then had equal amounts. How many dollars had each at first?