Lesson 2 · from §75 of the 1910 text
Dividing in a Given Ratio
Dividing in a given ratio
To separate a number into parts in the ratio 2 : 3, call the parts 2x and 3x. Their ratio is automatically 2 : 3.
Example. Separate 60 into three parts which are to each other as 3 : 4 : 5.
Let the parts be 3x, 4x, and 5x. Then 3x + 4x + 5x = 60, so 12x = 60 and x = 5. The parts are 15, 20, and 25.
Changing both terms
Example. What number added to both terms of \frac35 gives \frac57? \frac{3 + n}{5 + n} = \frac57, \quad 21 + 7n = 25 + 5n, \quad n = 2.
Comparing ratios
To tell which of two ratios is greater, write them with a common denominator and compare the numerators. For instance, \frac{a + 1}{a + 2} and \frac{a + 2}{a + 3} become \frac{(a + 1)(a + 3)}{(a + 2)(a + 3)} = \frac{a^2 + 4a + 3}{\cdots} \quad\text{and}\quad \frac{(a + 2)^2}{(a + 2)(a + 3)} = \frac{a^2 + 4a + 4}{\cdots}, so for positive a the second is greater.
Try one:
Quick check
Separate 60 into two parts which are in the ratio of 1 : 4.
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.
Give numbers only.
Separate 40 into two parts which are in the ratio of 2 : 3.
Separate 16 into two parts which are in the ratio of 5 : 3.
Separate 84 into two parts which are in the ratio of 3 : 11.
Separate 36 into three parts which are to each other as 2 : 3 : 4.
Separate 135 into three parts which are to each other as 4 : 5 : 6.
What number added to both terms of the ratio \frac58 gives as the result the ratio \frac{21}{28}?
What number subtracted from both terms of the ratio \frac{13}{27} gives as the result the ratio \frac29?
If a is a positive number, which is the greater ratio, \dfrac{4 + 2a}{4 + 3a} or \dfrac{4 + 3a}{4 + 4a}? Type the greater one.
If a and b are positive numbers, which is the greater ratio, \dfrac{a + 4b}{a + 5b} or \dfrac{a + 6b}{a + 7b}? Type the greater one.
If a positive number is added to both terms of a proper fraction, is the numerical value of the fraction increased?