**High School Number Sense Lesson 79: Estimating with Multiplication by 142,857**

You may recognize the number 142857, which is the order of the repeating decimal equivalent of

**1/7**. A fun estimation concept that shows up on high school tests is multiplying by this number, or by some other combination of the same digits (in the same order). This question appeared

**10 times**last year, with a median placement at

**question # 80**.

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The number 142,857 can be approximated as 1/7 of 1,000,000. (It is actually 1/7 of 999,999--but we're obviously close enough). So any time you see this number in an estimation question, simply divide the other number by 7 and then add 6 zeros.

Sometimes the number being multiplied has the same digits in the same order, but beginning with something other than 1. Just remember this pattern:

**1/7 = .142857...**

2/7 = .285714...

3/7 = .428571...

4/7 = .571428...

5/7 = .714285...

6/7 = .857142...

2/7 = .285714...

3/7 = .428571...

4/7 = .571428...

5/7 = .714285...

6/7 = .857142...

**Example 1: * 142857 x 36 = ___**

- 142857 is about 1/7 of 1,000,000. Divide 36 by 7 and then multiply by 1,000,000.
- 36 ÷ 7 =
**5 1/7**. 5 1/7 x 1,000,000 is about**5142857**. (The exact answer is**5142852**).

**Example 2: * 142.857 x 293 = ___**

- 142.857 is about 1/7 of 1000. Divide 293 by 7 and then multiply by 1000.
- 293 ÷ 7 =
**41 6/7**. 41 6/7 x 1000 is about**42000**. (The exact answer is**41857.101**).

**Example 3: * 571428 x 148 = ___**

- 571428 is about 4/7 of 1,000,000. Divide 148 by 7, multiply by 4, and then multiply by 1,000,000.
- 148 ÷ 7 =
**21 1/7**. 21 1/7 x 4 =**84 4/7**. - 84 4/7 x 1,000,000 is about
**84571428**. (The exact answer is**84571344**).

**Example 4: * 5714.28 x 63 = ___**

- 5714.28 is about 4/7 of 10,000. Divide 63 by 7, multiply by 4, and then multiply by 10,000.
- 63 ÷ 7 =
**9**. 9 x 4 =**36**. 36 x 10,000 =**360,000**. (The exact answer is**359999.64**).

**Example 5: * 286 x 357 = ___**

- 286 is about 2/7 of 1000. Divide 357 by 7, multiply by 2, and then multiply by 1000.
- 357 ÷ 7 =
**51**. 51 x 2 =**102**. 102 x 1000 =**102000**. (The exact answer is**102102**).

**Example 6: * 315 x 1428 = ___**

- 1428 is about 1/7 of 10,000. Divide 315 by 7 and then multiply by 10,000.
- 315 ÷ 7 =
**45**. 45 x 10,000 =**450000**. (The exact answer is**449820**).

**Check back soon for a free worksheet to help you practice EstMult142857.**

**Up Next for High School: EstMoneyInter**