Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest?

• $\frac{15}{11}<\frac{17}{13}<\frac{19}{15}$
• $\frac{15}{11}<\frac{19}{15}<\frac{17}{13}$
• $\frac{17}{13}<\frac{19}{15}<\frac{15}{11}$
• $\frac{19}{15}<\frac{15}{11}<\frac{17}{13}$
• $\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

AMC 8 2019 Problem 3

$\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$

First try to make all the denominators equal

$\frac{15}{11}, \frac{19}{15}, \frac{17}{13}$

=$\frac{15 \cdot 15 \cdot 13}{11 \cdot 15 \cdot 13}, \frac{19 \cdot 11 \cdot 13}{15 \cdot 11 \cdot 13}, \frac{17 \cdot 11 \cdot 15}{13 \cdot 11 \cdot 15}$

=$\frac{2925}{2145}, \frac{2717}{2145}, \frac{2805}{2145}$

Now, Try compare numerator

We have, $2717<2805<2925$

so , $\frac{19}{15}<\frac{17}{13}<\frac{15}{11}$ is the right order

Subscribe to Cheenta at Youtube

Ike and Mike go into a sandwich shop with a total of $\$ 30.00$ to spend. Sandwiches cost $\$ 4.50$ each and soft drinks cost $\$ 1.00$ each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?

• 6
• 7
• 8
• 9
• 10

AMC 8 2019 Problem 1

9

Try to start with

Let $s$ be the number of sandwiches and $d$ be the number of sodas. So it we will have
$$
4.50 s+d=30
$$

Now look , Ike and Mike buys maxixmum number of sandwitch possible, we can say $4.50s=30$ but s is integrer so the maximum s can be is 6 that is $4.50 \times 6 = 27$ So, $\$ 3.00$ is remaining.

So, the number if sodas is 3,

So, The number of items will be,

$6+3=9$

Subscribe to Cheenta at Youtube

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7$, the machine will output $3 \cdot 7+1=22$. Then if the output is repeatedly inserted into the machine five more times, the final output is 26.

$7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52\\ \rightarrow 26$

When the same 6 -step process is applied to a different starting value of $N$, the final output is 1 . What is the sum of all such integers $N$ ?

$N \rightarrow \longrightarrow \rightarrow-\rightarrow \longrightarrow \rightarrow 1$

• 73
• 74
• 75
• 82
• 83

AMC 8 2020 Problem 22

83

Try to start with the final output and work backwards.

Try to form a tree keeping in mind all the possible outcomes.

So, the sum will be,

$1+8+64+10=83$

Subscribe to Cheenta at Youtube

A small zoo has a giraffe, an elephant, a lion and a turtle. Susi wants to visit exactly two of the animals today but does not want to start with the lion. How many different possibilities does she have, to visit the two animals one after the other?

• 3
• 7
• 8
• 9
• 12

Math Kangaroo (Ecolier) 2017 Problem 22

9

Try to organize the choices to make sure that you count each and every option only once.

Let us try to consider the case where she start to visit from Giraffe.

So then we can have 3 ways to visit i.e. Giraffe, Lion; Giraffe, Turtle; Giraffe, Elephant.

In this way try to find out all the other ways we can have for visiting the zoo.

Also we have to remember that we can't start with Lion.

Other ways are elephant, lion; elephant, turtle; elephant, giraffe and turtle, giraffe; turtle, elephant; turtle, lion.

So in total we can have 3+3+3= 9 ways.

Subscribe to Cheenta at Youtube

How many integers between 2020 and 2400 have four distinct digits arranged in increasing order? (For example, 2347 is one integer.

• 9
• 10
• 15
• 21
• 28

AMC 8 2020 Problem 7

15

The second digit can't be 1 or 2, since the digit need to be increasing and distinct , and the second digit can't be 4 also since the number need to be less than 2400, so its 3

now we need to choose the last two digit from the set $\{4,5,6,7,8,9\}$

now we can do it in $6C2= 15$ ways. now in only one way we can order so there are 15 numbers.

Subscribe to Cheenta at Youtube