1 6 + 1 12 + 1 20 + 1 30 + 1 42 + 1 56 + 1 72 = ? { \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} = ? } 6 1 ​ + 12 1 ​ + 20 1 ​ + 30 1 ​ + 42 1 ​ + 56 1 ​ + 72 1 ​ = ?

by ADMIN 224 views

Introduction

In this article, we will delve into the world of mathematics and explore a fascinating problem that involves adding fractions with different denominators. The problem is as follows: 16+112+120+130+142+156+172=?{ \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} = ? }. We will break down the solution into manageable steps, making it easy to understand and follow along.

Understanding the Problem

Before we begin, let's take a closer look at the problem. We are given seven fractions with different denominators, and we need to find their sum. The fractions are:

  • 16{ \frac{1}{6} }
  • 112{ \frac{1}{12} }
  • 120{ \frac{1}{20} }
  • 130{ \frac{1}{30} }
  • 142{ \frac{1}{42} }
  • 156{ \frac{1}{56} }
  • 172{ \frac{1}{72} }

Step 1: Finding the Least Common Multiple (LCM)

To add fractions with different denominators, we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of all the denominators. In this case, the denominators are 6, 12, 20, 30, 42, 56, and 72.

To find the LCM, we can list the multiples of each denominator and find the smallest number that appears in all the lists.

  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
  • Multiples of 12: 12, 24, 36, 48, 60, 72, ...
  • Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, 700, 720, ...
  • Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, ...
  • Multiples of 42: 42, 84, 126, 168, 210, 252, 294, 336, 378, 420, 462, 504, 546, 588, 630, 672, 714, 756, 798, 840, 882, 924, 966, 1008, 1050, 1092, 1134, 1176, 1218, 1260, 1302, 1344, 1386, 1428, 1470, 1512, 1554, 1596, 1638, 1680, 1722, 1764, 1806, 1848, 1890, 1932, 1974, 2016, 2058, 2100, 2142, 2184, 2226, 2268, 2310, 2352, 2394, 2436, 2478, 2520, 2562, 2604, 2646, 2688, 2730, 2772, 2814, 2856, 2898, 2940, 2982, 3024, 3066, 3108, 3150, 3192, 3234, 3276, 3318, 3360, 3402, 3444, 3486, 3528, 3570, 3612, 3654, 3696, 3738, 3780, 3822, 3864, 3906, 3948, 3990, 4032, 4074, 4116, 4158, 4200, 4242, 4284, 4326, 4368, 4410, 4452, 4494, 4536, 4578, 4620, 4662, 4704, 4746, 4788, 4830, 4872, 4914, 4956, 4998, 5040, 5082, 5124, 5166, 5208, 5250, 5292, 5334, 5376, 5418, 5460, 5502, 5544, 5586, 5628, 5670, 5712, 5754, 5796, 5838, 5880, 5922, 5964, 6006, 6048, 6090, 6132, 6174, 6216, 6258, 6300, 6342, 6384, 6426, 6468, 6510, 6552, 6594, 6636, 6678, 6720, 6762, 6804, 6846, 6888, 6930, 6972, 7014, 7056, 7098, 7140, 7182, 7224, 7266, 7308, 7350, 7392, 7434, 7476, 7518, 7560, 7602, 7644, 7686, 7728, 7770, 7812, 7854, 7896, 7938, 7980, 8022, 8064, 8106, 8148, 8190, 8232, 8274, 8316, 8358, 8400, 8442, 8484, 8526, 8568, 8610, 8652, 8694, 8736, 8778, 8820, 8862, 8904, 8946, 8988, 9030, 9072, 9114, 9156, 9198, 9240, 9282, 9324, 9366, 9408, 9450, 9492, 9534, 9576, 9618, 9660, 9702, 9744, 9786, 9828, 9870, 9912, 9954, 9996, 10038, 10080, 10122, 10164, 10206, 10248, 10290, 10332, 10374, 10416, 10458, 10500, 10542, 10584, 10626, 10668, 10710, 10752, 10794, 10836, 10878, 10920, 10962, 11004, 11046, 11088, 11130, 11172, 11214, 11256, 11298, 11340, 11382, 11424, 11466, 11508, 11550, 11592, 11634, 11676, 11718, 11760, 11802, 11844, 11886, 11928, 11970, 12012, 12054, 12096, 12138, 12180, 12222, 12264, 12306, 12348, 12390, 12432, 12474, 12516, 12558, 12600, 12642, 12684, 12726, 12768, 12810, 12852, 12894, 12936, 12978, 13020, 13062, 13104, 13146, 13188, 13230, 13272, 13314, 13356, 13400, 13442, 13484, 13526, 13568, 13610, 13652, 13694, 13736, 13778, 13820, 13862, 13904, 13946, 13988, 14030, 14072, 14114, 14156, 14200, 14242, 14284, 14326, 14368, 14410, 14452, 14494, 14536, 14578, 14620, 14662, 14704, 14746, 14788, 14830, 14872, 14914,
    Solving the Puzzle: A Step-by-Step Guide to Adding Fractions with Different Denominators ===========================================================

Q&A: Frequently Asked Questions

Q: What is the least common multiple (LCM) and why is it important in adding fractions with different denominators? A: The least common multiple (LCM) is the smallest number that is a multiple of all the denominators. It is important in adding fractions with different denominators because it allows us to find a common denominator for all the fractions, making it easier to add them.

Q: How do I find the LCM of a set of numbers? A: To find the LCM of a set of numbers, you can list the multiples of each number and find the smallest number that appears in all the lists. Alternatively, you can use the prime factorization method, which involves finding the prime factors of each number and multiplying them together to get the LCM.

Q: What is the prime factorization method and how does it work? A: The prime factorization method involves breaking down each number into its prime factors and then multiplying them together to get the LCM. For example, if we want to find the LCM of 6 and 12, we can break them down into their prime factors: 6 = 2 x 3 and 12 = 2^2 x 3. Then, we multiply the prime factors together to get the LCM: 2^2 x 3 = 12.

Q: How do I add fractions with different denominators using the LCM? A: To add fractions with different denominators using the LCM, you need to follow these steps:

  1. Find the LCM of the denominators.
  2. Convert each fraction to have the LCM as the denominator.
  3. Add the fractions together.

For example, if we want to add 1/6 and 1/12, we can follow these steps:

  1. Find the LCM of 6 and 12, which is 12.
  2. Convert 1/6 to have a denominator of 12: 1/6 = 2/12.
  3. Add the fractions together: 2/12 + 1/12 = 3/12.

Q: What if the LCM is not a whole number? A: If the LCM is not a whole number, you can simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD). For example, if the LCM is 12.5, you can simplify the fraction by dividing the numerator and denominator by 2.5 to get 5/5, which is equal to 1.

Q: Can I use a calculator to find the LCM and add fractions with different denominators? A: Yes, you can use a calculator to find the LCM and add fractions with different denominators. Many calculators have a built-in function to find the LCM and add fractions with different denominators.

Conclusion

Adding fractions with different denominators can be a challenging task, but with the right tools and techniques, it can be made easier. By finding the LCM of the denominators and converting each fraction to have the LCM as the denominator, you can add fractions with different denominators with ease. Remember to simplify the fraction by dividing the numerator and denominator by GCD if the LCM is not a whole number. With practice and patience, you can become proficient in adding fractions with different denominators.

Final Answer

Now that we have gone through the steps to add fractions with different denominators, let's go back to the original problem: 16+112+120+130+142+156+172=?{ \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} = ? }.

Using the steps we learned, we can find the LCM of the denominators, which is 840. Then, we can convert each fraction to have a denominator of 840 and add them together:

16=140840{ \frac{1}{6} = \frac{140}{840} }

112=70840{ \frac{1}{12} = \frac{70}{840} }

120=42840{ \frac{1}{20} = \frac{42}{840} }

130=28840{ \frac{1}{30} = \frac{28}{840} }

142=20840{ \frac{1}{42} = \frac{20}{840} }

156=15840{ \frac{1}{56} = \frac{15}{840} }

172=11.67840{ \frac{1}{72} = \frac{11.67}{840} } (approximately)

Adding these fractions together, we get:

140840+70840+42840+28840+20840+15840+11.67840≈326.67840{ \frac{140}{840} + \frac{70}{840} + \frac{42}{840} + \frac{28}{840} + \frac{20}{840} + \frac{15}{840} + \frac{11.67}{840} \approx \frac{326.67}{840} }

Simplifying this fraction, we get:

326.67840≈0.39{ \frac{326.67}{840} \approx 0.39 }

Therefore, the final answer is approximately 0.39.