Convert each of the following repeating decimals to a/b form, where a, b are integers and b ≠ 0.
#2a ¯
0.4
#2c ¯
1.396
#2e ¯
-2.34
2a.
0.4̅
=4/9
The numerator is obtained by moving the decimal point one place to the right.
The denominator is the number of "9" digits that correspond to the number of decimal places moved.
To convert a repeating decimal to a fraction in the form of a/b, we need to follow these steps:
Step 1: Identify the pattern of the repeating decimal.
Step 2: Assign a variable to the repeating part of the decimal.
Step 3: Subtract the repeating part from the original number to eliminate the repeating part.
Step 4: Multiply both sides of the equation from step 3 by an appropriate power of 10 to eliminate the decimal point.
Step 5: Solve for the variable from step 2.
Step 6: Write the fraction in the form of a/b.
Now let's apply these steps to each of the given repeating decimals:
#2a) 0.4¯
Step 1: The pattern here is "4" repeating.
Step 2: Let x = 0.44¯
Step 3: 10x = 4.4¯
Subtracting x from 10x, we get:
10x - x = 4.4¯ - 0.4¯
Simplifying:
9x = 4
x = 4/9
Step 4: Multiply both sides by 10 to eliminate the decimal point:
10x = 4
10 * (4/9) = 4
40/9 = 4
Step 5: We have found x = 4/9
Step 6: The repeating decimal 0.4¯ can be written as the fraction 4/9.
#2c) 1.396¯
Step 1: The pattern here is "396" repeating.
Step 2: Let x = 1.396¯
Step 3: 100x = 139.6¯
Subtracting x from 100x, we get:
100x - x = 139.6¯ - 1.396¯
Simplifying:
99x = 138.2
x = 138.2/99
Step 4: Multiply both sides by 100 to eliminate the decimal point:
100x = 138.2
100 * (138.2/99) = 138.2
13820/99 = 138.2
Step 5: We have found x = 13820/99
Step 6: The repeating decimal 1.396¯ can be written as the fraction 13820/99.
#2e) -2.34¯
Step 1: The pattern here is "34" repeating.
Step 2: Let x = -2.3434¯
Step 3: 100x = -234.3434¯
Subtracting x from 100x, we get:
100x - x = -234.3434¯ - (-2.34¯)
Simplifying:
99x = -231
Step 4: Multiply both sides by 100 to eliminate the decimal point:
100x = -231
100 * (-231/99) = -231
-23100/99 = -231
Step 5: We have found x = -23100/99
Step 6: The repeating decimal -2.34¯ can be written as the fraction -23100/99.
2c
2.34̅
First, consider only the decimal part as an integer, i=34.
Count the number of digits that repeat (x=1) and those that don't (y=1).
From i=34, strip of the part that repeats to get j=3
Subtract j from i to get the numerator=34-3=31.
The numerator is obtained by stringing together x 9's and y 0's, to get 90.
The fraction is then 31/90.
Add to the integral part (2) to get
2 31/90. Transform the number to the required form as required.
correction:
The denominator is obtained by...