Convert the recurring decimal 0.7̇ to a fraction in its simplest form.
Worked Solution & Answer
Let x = 0.777…, so 10x = 7.777… and 9x = 7 ⇒ x = 79 B1
Answer: 79
Recurring decimals — Notes & Worked Examples Show / Hide
Explanation:
To convert a recurring decimal into a fraction algebraically, let x equal the recurring decimal written out with several repeating digits. Multiply x by powers of 10 (10x, 100x, 1000x) so that two equations have the exact same recurring tail after the decimal point.
Subtract the smaller equation from the larger equation so that all the recurring digits after the decimal point cancel out to zero, then divide and simplify the fraction.
Example 1: Prove algebraically that 0.̇4̇5 = 511.
Solution:
Let x = 0.454545...
Multiply by 100 (2 repeating digits): 100x = 45.454545...
Subtract (100x − x): 99x = 45
x = 4599 = 511 (divide top and bottom by 9)
Answer: 511
Example 2: Convert 0.2̇7̇3 (where 73 recurs) into a fraction in its simplest form.
Solution:
Let x = 0.2737373... → 10x = 2.737373...
Multiply by 1000: 1000x = 273.737373...
Subtract (1000x − 10x): 990x = 271
Divide by 990: x = 271990
Answer: 271990
Convert the recurring decimal 0.4̇ to a fraction in its simplest form.
Worked Solution & Answer
Let x = 0.444…, so 10x = 4.444… and 9x = 4 ⇒ x = 49 B1
Answer: 49
Using algebra, prove that 1.062̇ can be written as 114225
Worked Solution & Answer
1000x – 100x = 956 M2
⇒ x = 956/900 = 239/225 = 1 14/225 C1
⇒ x = 956/900 = 239/225 = 1 14/225 C1
Answer: Let x = 1.06222..., 100x = 106.222..., 1000x = 1062.222...; 900x = 956 ⇒ x = 956/900 = 1 56/900 = 1 14/225
Express 0.11̇7̇ as a fraction.
You must show all your working.
You must show all your working.
Worked Solution & Answer
Let x = 0.1171717... M1
10x = 1.171717... and 1000x = 117.171717... M1
990x = 116 ⇒ x = 116/990 (or 58/495) A1
10x = 1.171717... and 1000x = 117.171717... M1
990x = 116 ⇒ x = 116/990 (or 58/495) A1
Answer: 116990
Prove algebraically that 0.07̇23̇ can be written as 2413330
Worked Solution & Answer
10000x – 10x = 723 M1
⇒ 9990x = 723 M1
⇒ x = 723/9990 = 241/3330 C1
⇒ 9990x = 723 M1
⇒ x = 723/9990 = 241/3330 C1
Answer: Let x = 0.0723723... ⇒ 10x = 0.723723... and 10000x = 723.723723... ⇒ 9990x = 723 ⇒ x = 723/9990 = 241/3330
x = 0.43̇6̇
Prove algebraically that x can be written as 2455
Prove algebraically that x can be written as 2455
Worked Solution & Answer
10x = 4.3636... and 1000x = 436.3636.. M2
Subtracting gives 990x = 432 ⇒ x = 432/990 = 24/55. C1
Subtracting gives 990x = 432 ⇒ x = 432/990 = 24/55. C1
Answer: 10x = 4.3̇6̇, 1000x = 436.3̇6̇ ⇒ 990x = 432 ⇒ x = 432990 = 2455
Prove algebraically that 0.25̇6̇ can be written as 127495
Worked Solution & Answer
x = 0.25656... ⇒ 10x = 2.5656..., 1000x = 256.5656.. M2
Subtracting gives 990x = 254 ⇒ x = 254/990 = 127/495. C1
Subtracting gives 990x = 254 ⇒ x = 254/990 = 127/495. C1
Answer: Let x = 0.25656...; 10x = 2.5656... and 1000x = 256.5656...; 990x = 254 ⇒ x = 254/990 = 127/495
Show that 0.1̇5̇ + 0.22̇7̇ can be written in the form m66 where m is an integer.
Worked Solution & Answer
0.1̇5̇ = 15/99 = 5/33 = 10/66 M1
Let y = 0.22̇7̇ ⇒ 10y = 2.2̇7̇ and 1000y = 227.2̇7̇ ⇒ 990y = 225 ⇒ y = 225/990 = 15/66 M1
Sum = 10/66 + 15/66 = 25/66. C1
Let y = 0.22̇7̇ ⇒ 10y = 2.2̇7̇ and 1000y = 227.2̇7̇ ⇒ 990y = 225 ⇒ y = 225/990 = 15/66 M1
Sum = 10/66 + 15/66 = 25/66. C1
Answer: 2566
The diagram shows a number line.
P = 1.2̇ and Q = 123.
Q is the midpoint of PR.
Find the value of R.
Give your answer as a mixed number in its simplest form.
You must show your working.
Q is the midpoint of PR.
Find the value of R.
Give your answer as a mixed number in its simplest form.
You must show your working.
Worked Solution & Answer
P = 1.2̇ = 1 2/9 = 11/9. M2
Q = 1 2/3 = 1 6/9 = 15/9. M1
Step from P to Q = 15/9 − 11/9 = 4/9. M1
R = Q + 4/9 = 15/9 + 4/9 = 19/9 = 2 1/9. A1
Q = 1 2/3 = 1 6/9 = 15/9. M1
Step from P to Q = 15/9 − 11/9 = 4/9. M1
R = Q + 4/9 = 15/9 + 4/9 = 19/9 = 2 1/9. A1
Answer: 219
x = 0.2̇, y = 0.68̇1̇
Work out the value of xy.
Give your answer as a fraction in its simplest form.
Give your answer as a fraction in its simplest form.
Worked Solution & Answer
x = 0.222..., 10x = 2.222... ⇒ 9x = 2 ⇒ x = 2/9. M2
y = 0.68181..., 10y = 6.8181..., 1000y = 681.8181... ⇒ 990y = 675 ⇒ y = 675/990 = 15/22. M2
xy = (2/9) × (15/22) = 30/198 = 5/33. A1
y = 0.68181..., 10y = 6.8181..., 1000y = 681.8181... ⇒ 990y = 675 ⇒ y = 675/990 = 15/22. M2
xy = (2/9) × (15/22) = 30/198 = 5/33. A1
Answer: 533