Daily Practice

Solution

LCM(6,8)=24. We need LCM(24,x)=24, which means x must be a divisor of 24. Among the choices, 4 divides 24. So x=4 works.

Solution

In order to maximize the sum of the numbers, the numbers must have their digits ordered in decreasing value. There are only two numbers from the answer choices with this property: $76531$ and $87431$. To determine the answer we will have to use estimation and the first two digits of the numbers. For $76531$ the number that would maximize the sum would start with $98$. The first two digits of $76531$ (when rounded) are $77$. Adding $98$ and $77$, we find that the first three digits of the sum of the two numbers would be $175$. For $87431$ the number that would maximize the sum would start with $96$. The first two digits of $87431$ (when rounded) are $87$. Adding $96$ and $87$, we find that the first three digits of the sum of the two numbers would be $183$. From the estimations, we can say that the answer to this problem is $boxed\{textbf\{(C)\}\; 87431\}$. p.s. USE INTUITION, see answer choices before solving any question -litttle_master == Solution 2 == In order to determine the largest number possible, we have to evenly distribute the digits when adding. The two numbers that show an example of this are $97531$ and $86420$. The digits can be interchangeable between numbers because we only care about the actual digits. The first digit must be either $9$ or $8$. This immediately knocks out $textbf\{(A)\}\; 76531$. The second digit must be either $7$ or $6$. This doesn't cancel any choices. The third digit must be either $5$ or $4$. This knocks out $textbf\{(B)\}\; 86724$ and $textbf\{(D)\}\; 96240$. The fourth digit must be $3$ or $2$. This cancels out $textbf\{(E)\}\; 97403$. This leaves us with $boxed\{textbf\{(C)\}\; 87431\}$. == Solution 3 == If we use intuition, we know that the digits will be IN ORDER, to maximze the number. This eliminates textbf{(B)}, textbf{(D)},and textbf{(E)}. Additionally, the 2 two numbers must have 9 or 8 for the first digit to maximize the sum, eliminating textbf{(A)}. This leaves boxed{textbf{(C)} 87431}. -written by litttle_master purely, not copied from anywhere == Video Solution by OmegaLearn== https://youtu.be/HISL2-N5NVg?t=654 ~ pi_is_3.14 https://youtu.be/trAjltkbSWo ~savannahsolver

Solution

After Gilda gives $20$% of the marbles to Pedro, she has $80$% of the marbles left. If she then gives $10$% of what's left to Ebony, she has $(0.8*0.9)$ = $72$% of what she had at the beginning. Finally, she gives $25$% of what's left to her brother, so she has $(0.75*0.72)$ $boxed\{textbf\{(E)\}\; 54\}$ of what she had in the beginning left.

Solution

We group the addends inside the parentheses two at a time: begin{align*} -1 + 2 - 3 + 4 - 5 + 6 - 7 + ldots + 1000 &= (-1 + 2) + (-3 + 4) + (-5 + 6) + ldots + (-999 + 1000) \ &= underbrace{1+1+1+ldots + 1}_{text{500 1's}} \ &= 500. end{align*} Then the desired answer is $4\; times\; 500\; =\; boxed\{textbf\{(E)\}\; 2000\}$. ==Video Solution== https://youtu.be/4Gy1CrYTDHA ~savannahsolver ==Video Solution by OmegaLearn== https://youtu.be/TkZvMa30Juo?t=3074 ~ pi_is_3.14

Solution

When M is first reflected over the line $q,$ we obtain the following diagram: [figure] When M is then reflected over the line $p,$ we obtain the following diagram: [figure] Therefore, the answer is $boxed\{textbf\{(E)\}\}.$ ~MRENTHUSIASM ==Video Solution (A Clever Explanation You’ll Get Instantly)== https://youtu.be/tYWp6fcUAik?si=V8hv_zOn_zYOi9E5&t=350 ~hsnacademy ==Video Solution 1 by Math-X (First understand the problem!!!)== https://youtu.be/oUEa7AjMF2A?si=l-XMI73aofM6q4ye&t=320 ~Math-X ==Video Solution 2 (CREATIVE THINKING!!!)== https://youtu.be/lItyKxX_qB8 ~Education, the Study of Everything ==Video Solution 3== https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=259 ~Interstigation ==Video Solution 4== https://youtu.be/c-D-bj2htGE ~savannahsolver ==Video Solution 5== https://youtu.be/Q0R6dnIO95Y?t=182 ~STEMbreezy ==Video Solution 6== https://www.youtube.com/watch?v=6kkJVsOKcuk ~harungurcan ==Video Solution 7 by Dr. David== https://youtu.be/hh3DbQ9Rd5Y