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Sagot :
Alright, let's solve each of the problems step-by-step.
### 1. What is [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] minus [tex]\(\frac{2}{5} \, \text{mm}\)[/tex]? Express the answer in terms of mm.
First, we need to convert [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] to mm.
- [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] is equal to [tex]\(5 + \frac{3}{10} \, \text{cm}\)[/tex].
- [tex]\(5 + \frac{3}{10} = 5.3 \, \text{cm}\)[/tex].
- Since 1 cm = 10 mm, [tex]\(5.3 \, \text{cm} = 5.3 \times 10 = 53 \, \text{mm}\)[/tex].
Next, we subtract [tex]\(\frac{2}{5} \, \text{mm}\)[/tex]:
- [tex]\(\frac{2}{5} = 0.4 \, \text{mm}\)[/tex].
Therefore,
[tex]\[53 \, \text{mm} - 0.4 \, \text{mm} = 52.6 \, \text{mm}.\][/tex]
So, the answer is [tex]\(52.6 \, \text{mm}\)[/tex].
### 2. What is the value of [tex]\(\left(\frac{1}{2} + 2 - \frac{1}{3}\right)\)[/tex]?
To find this, we need to perform the arithmetic operations step-by-step.
- [tex]\(\frac{1}{2} = 0.5\)[/tex],
- [tex]\(2 = 2\)[/tex],
- [tex]\(\frac{1}{3} \approx 0.333\)[/tex].
Now calculate:
[tex]\[0.5 + 2 - 0.333 = 2.1666666666666665.\][/tex]
Thus, [tex]\(\left(\frac{1}{2} + 2 - \frac{1}{3}\right) = 2.1666666666666665\)[/tex].
### 3. Luigi spent [tex]\(\frac{4}{5}\)[/tex] of his money on books. He spent another [tex]\(\frac{1}{7}\)[/tex] of his money on pens. What fraction of his money was left?
First, let's determine how much of his money he spent.
- He spent [tex]\(\frac{4}{5}\)[/tex] of his money on books.
- After spending on books, he has [tex]\(\frac{1}{5}\)[/tex] of his money left.
- He then spent [tex]\(\frac{1}{7}\)[/tex] of his remaining money (which is [tex]\(\frac{1}{5}\)[/tex]) on pens.
So, the amount spent on pens is:
[tex]\[ \frac{1}{7} \times \frac{1}{5} = \frac{1}{35}. \][/tex]
Total spent:
[tex]\[ \frac{4}{5} + \frac{1}{35} = \frac{28}{35} + \frac{1}{35} = \frac{29}{35}. \][/tex]
The fraction of money left:
[tex]\[ 1 - \frac{29}{35} = \frac{35}{35} - \frac{29}{35} = \frac{6}{35} \approx 0.17142857142857137. \][/tex]
So, Luigi has approximately 0.17142857142857137 (or [tex]\(\frac{6}{35}\)[/tex]) of his money left.
### 4. Mrs. Baclaya made some muffins and gave them to Vincent and Brylle. Vincent received [tex]\(\frac{1}{4}\)[/tex] of the number of muffins, and Brylle received [tex]\(\frac{2}{3}\)[/tex] of the remainder. How many muffins did Mrs. Baclaya make if she had 9 muffins left?
Let [tex]\(M\)[/tex] be the total number of muffins.
- Vincent received [tex]\(\frac{1}{4}M\)[/tex]. Therefore, Mrs. Baclaya had [tex]\(M - \frac{1}{4}M = \frac{3}{4}M\)[/tex] muffins left.
- Brylle received [tex]\(\frac{2}{3}\)[/tex] of the remaining [tex]\(\frac{3}{4}M\)[/tex].
Brylle received:
[tex]\[ \frac{2}{3} \times \frac{3}{4}M = \frac{2}{4}M = \frac{1}{2}M. \][/tex]
So, the muffins left after Brylle received hers are:
[tex]\[ \frac{3}{4}M - \frac{1}{2}M = \frac{3M}{4} - \frac{2M}{4} = \frac{M}{4}. \][/tex]
Given that [tex]\(\frac{M}{4} = 9\)[/tex]:
[tex]\[ \frac{M}{4} = 9 \Rightarrow M = 9 \times 4 = 36. \][/tex]
Therefore, Mrs. Baclaya made 36 muffins.
To summarize:
1. [tex]\(52.6 \, \text{mm}\)[/tex]
2. [tex]\(2.1666666666666665\)[/tex]
3. [tex]\(0.17142857142857137\)[/tex]
4. [tex]\(36\)[/tex] muffins
### 1. What is [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] minus [tex]\(\frac{2}{5} \, \text{mm}\)[/tex]? Express the answer in terms of mm.
First, we need to convert [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] to mm.
- [tex]\(5 \frac{3}{10} \, \text{cm}\)[/tex] is equal to [tex]\(5 + \frac{3}{10} \, \text{cm}\)[/tex].
- [tex]\(5 + \frac{3}{10} = 5.3 \, \text{cm}\)[/tex].
- Since 1 cm = 10 mm, [tex]\(5.3 \, \text{cm} = 5.3 \times 10 = 53 \, \text{mm}\)[/tex].
Next, we subtract [tex]\(\frac{2}{5} \, \text{mm}\)[/tex]:
- [tex]\(\frac{2}{5} = 0.4 \, \text{mm}\)[/tex].
Therefore,
[tex]\[53 \, \text{mm} - 0.4 \, \text{mm} = 52.6 \, \text{mm}.\][/tex]
So, the answer is [tex]\(52.6 \, \text{mm}\)[/tex].
### 2. What is the value of [tex]\(\left(\frac{1}{2} + 2 - \frac{1}{3}\right)\)[/tex]?
To find this, we need to perform the arithmetic operations step-by-step.
- [tex]\(\frac{1}{2} = 0.5\)[/tex],
- [tex]\(2 = 2\)[/tex],
- [tex]\(\frac{1}{3} \approx 0.333\)[/tex].
Now calculate:
[tex]\[0.5 + 2 - 0.333 = 2.1666666666666665.\][/tex]
Thus, [tex]\(\left(\frac{1}{2} + 2 - \frac{1}{3}\right) = 2.1666666666666665\)[/tex].
### 3. Luigi spent [tex]\(\frac{4}{5}\)[/tex] of his money on books. He spent another [tex]\(\frac{1}{7}\)[/tex] of his money on pens. What fraction of his money was left?
First, let's determine how much of his money he spent.
- He spent [tex]\(\frac{4}{5}\)[/tex] of his money on books.
- After spending on books, he has [tex]\(\frac{1}{5}\)[/tex] of his money left.
- He then spent [tex]\(\frac{1}{7}\)[/tex] of his remaining money (which is [tex]\(\frac{1}{5}\)[/tex]) on pens.
So, the amount spent on pens is:
[tex]\[ \frac{1}{7} \times \frac{1}{5} = \frac{1}{35}. \][/tex]
Total spent:
[tex]\[ \frac{4}{5} + \frac{1}{35} = \frac{28}{35} + \frac{1}{35} = \frac{29}{35}. \][/tex]
The fraction of money left:
[tex]\[ 1 - \frac{29}{35} = \frac{35}{35} - \frac{29}{35} = \frac{6}{35} \approx 0.17142857142857137. \][/tex]
So, Luigi has approximately 0.17142857142857137 (or [tex]\(\frac{6}{35}\)[/tex]) of his money left.
### 4. Mrs. Baclaya made some muffins and gave them to Vincent and Brylle. Vincent received [tex]\(\frac{1}{4}\)[/tex] of the number of muffins, and Brylle received [tex]\(\frac{2}{3}\)[/tex] of the remainder. How many muffins did Mrs. Baclaya make if she had 9 muffins left?
Let [tex]\(M\)[/tex] be the total number of muffins.
- Vincent received [tex]\(\frac{1}{4}M\)[/tex]. Therefore, Mrs. Baclaya had [tex]\(M - \frac{1}{4}M = \frac{3}{4}M\)[/tex] muffins left.
- Brylle received [tex]\(\frac{2}{3}\)[/tex] of the remaining [tex]\(\frac{3}{4}M\)[/tex].
Brylle received:
[tex]\[ \frac{2}{3} \times \frac{3}{4}M = \frac{2}{4}M = \frac{1}{2}M. \][/tex]
So, the muffins left after Brylle received hers are:
[tex]\[ \frac{3}{4}M - \frac{1}{2}M = \frac{3M}{4} - \frac{2M}{4} = \frac{M}{4}. \][/tex]
Given that [tex]\(\frac{M}{4} = 9\)[/tex]:
[tex]\[ \frac{M}{4} = 9 \Rightarrow M = 9 \times 4 = 36. \][/tex]
Therefore, Mrs. Baclaya made 36 muffins.
To summarize:
1. [tex]\(52.6 \, \text{mm}\)[/tex]
2. [tex]\(2.1666666666666665\)[/tex]
3. [tex]\(0.17142857142857137\)[/tex]
4. [tex]\(36\)[/tex] muffins
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