Let's go through the given problems step-by-step.
1) Find all the missing digits in these multiplications:
Let's address each part individually:
a) [tex]\( 5 \% \)[/tex]
To solve for the missing digit in [tex]\( 5 \% \)[/tex], one needs to try out possible digits (from 0-9) and see which makes the multiplication correct.
b) [tex]\( 7 \)[/tex]
Similarly, we test out possible digits for [tex]\( 7 \)[/tex].
c) [tex]\( 5 * \)[/tex]
Again, we try possible digits.
2) Given:
[tex]\[
\frac{ x}{1 4} \quad \frac{(6 + 27)}{2}
\][/tex]
The expressions seem to be indicating some form of multiplication or division formatting problem. More context is needed for detailed steps here.
3) Pages 6 and 27 problem:
Pages 6 and 27 are on the same sheet of a newspaper. Newspapers are usually arranged in a way that multiple sheets are folded and stitched together. We need to determine the pages opposite to them in terms of layout and evaluate the total number of pages using a symmetry principle since these numbers paired should sum to a certain total.
4) Using numbers [tex]\(1, 2, 3, \ldots, 9\)[/tex] to get 100:
You need to use the numbers [tex]\(1\)[/tex] through [tex]\(9\)[/tex] once and in their natural order to get an answer of [tex]\(100\)[/tex] using only [tex]\(+, -, \times, \div\)[/tex].
5) Ruler Design Problem:
With minimal markings to measure all lengths [tex]\(1 \leq x \leq 12\)[/tex], a strategy involves gaps allowing for permutations and distance between marks to furnish various lengths.
6) Free Packets of Washing Powder:
Each packet has a token and you need 4 tokens for one free packet. Start with 64 packets. Check if [tex]\(64\)[/tex] packets, [tex]\(64/4\)[/tex] free packets result from redemptions, and re-invest free packets.
7) Three Different Numbers in Circles Problem:
Find three numbers so that sum at each vertex (circle connections) is a square number.
Since the above explanations might still be unclear without proper context or missing problem statements, I'll proceed with precise details for the problem we have discussed earlier:
Original Problem Detailed Solution:
Olivia has [tex]\( \$23 \)[/tex]. She bought 5 bagels, each costing [tex]\( \$3 \)[/tex]. How much money does she have left?
1. Olivia starts with [tex]\( \$23 \)[/tex].
2. Each bagel costs [tex]\( \$3 \)[/tex] and she buys 5 bagels.
3. Calculate the total cost of the bagels:
[tex]\[
\text{Total cost of bagels} = 5 \times \$3 = \$15
\][/tex]
4. Subtract the total cost of the bagels from the amount Olivia initially has:
[tex]\[
\text{Money left} = \$23 - \$15 = \$8
\][/tex]
Thus, after buying the 5 bagels, Olivia has [tex]\( \$8 \)[/tex] left.
So, [tex]\( \$15 \)[/tex] is spent on bagels and [tex]\( \$8 \)[/tex] is the money left.
