Math / Algebra

QuestionХоёр тооны нийлбэр 2024 их тоо бага тоонд хуваахад / ноогдож 16 үлдэнэ. Эдгээр 257 1773 202.17 Хэрэв тооны гуравны нэг дээр 20-г нэмэхэд анхны тооноос 251= 1713 Дахин их тоо гардаг бол уг тоог ол. Бутархайн хуртвэр хуваарийн нийлбэр 21 хуртвэрээс 2-ыг хасаж хуваарь дээр 1-ийг

Studdy Solution

STEP 1

What is this asking? We've got two separate puzzles here!
One involves two numbers that add up to 2024, with a specific remainder when dividing.
The other asks us to find a number based on how it relates to one-third of itself plus 20. Watch out! Don't mix up the two problems!
And remember, remainders mean we might have to do some clever subtracting.

STEP 2

1. Solve the division puzzle
2. Solve the one-third plus 20 puzzle

STEP 3

Alright, let's tackle the first mystery!
We know two numbers add up to 2024, and when we divide the bigger one by the smaller one, we get a remainder of 16.

STEP 4

Think of it like this: If it weren't for that pesky remainder, the bigger number would be a perfect multiple of the smaller number.
So, let's subtract that remainder! $\ 2024 - 16 = 2008$ .
Now, we know that $\ 2008$ is perfectly divisible by the smaller number.

STEP 5

We also know that the two numbers add up to $\ 2024$ .
Since we subtracted the $\ 16$ from the total, we need to subtract it from the larger number as well.
Let's call the smaller number $x$ .
Then, the larger number is $x + 16$ .
So, $x + (x + 16) = 2024$ .
Simplifying gives us $2x + 16 = 2024$ .

STEP 6

Subtract $16$ from both sides: $2x = 2024 - 16$ , which means $2x = 2008$ .
Now, divide both sides by $\ 2$ : $x = \frac{2008}{2}$ , so $x = 1004$ .
That's our smaller number!

STEP 7

The larger number is $\ 2024 - 1004 = 1020$ .
Let's check: $\ \frac{1020}{1004} = 1$ with a remainder of $\ 16$ .
Perfect!

STEP 8

Let's call our mystery number $y$ .
The problem says one-third of the number ( $\frac{1}{3} \cdot y$ ) plus $\ 20$ equals twice the original number ( $2 \cdot y$ ).
Let's write that as an equation: $\frac{1}{3} \cdot y + 20 = 2 \cdot y$ .

STEP 9

Let's get rid of the fraction!
Multiply everything by $\ 3$ : $3 \cdot (\frac{1}{3} \cdot y + 20) = 3 \cdot (2 \cdot y)$ .
This simplifies to $y + 60 = 6y$ .

STEP 10

Now, subtract $y$ from both sides: $60 = 6y - y$ , which means $60 = 5y$ .

STEP 11

Finally, divide both sides by $\ 5$ : $y = \frac{60}{5}$ , so $y = 12$ .
Our mystery number is $\ 12$ !

STEP 12

The two numbers that add up to 2024 and have a remainder of 16 when divided are 1004 and 1020.
The mystery number that becomes twice itself when you take one-third of it and add 20 is 12.

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Studdy Solution

STEP 1

STEP 2

1. Solve the division puzzle
2. Solve the one-third plus 20 puzzle

STEP 3

Alright, let's tackle the first mystery!
We know two numbers add up to 2024, and when we divide the bigger one by the smaller one, we get a remainder of 16.

STEP 4

Think of it like this: If it weren't for that pesky remainder, the bigger number would be a perfect multiple of the smaller number.
So, let's subtract that remainder! $\ 2024 - 16 = 2008$ .
Now, we know that $\ 2008$ is perfectly divisible by the smaller number.

STEP 5

We also know that the two numbers add up to $\ 2024$ .
Since we subtracted the $\ 16$ from the total, we need to subtract it from the larger number as well.
Let's call the smaller number $x$ .
Then, the larger number is $x + 16$ .
So, $x + (x + 16) = 2024$ .
Simplifying gives us $2x + 16 = 2024$ .

STEP 6

Subtract $16$ from both sides: $2x = 2024 - 16$ , which means $2x = 2008$ .
Now, divide both sides by $\ 2$ : $x = \frac{2008}{2}$ , so $x = 1004$ .
That's our smaller number!

STEP 7

The larger number is $\ 2024 - 1004 = 1020$ .
Let's check: $\ \frac{1020}{1004} = 1$ with a remainder of $\ 16$ .
Perfect!

STEP 8

Let's call our mystery number $y$ .
The problem says one-third of the number ( $\frac{1}{3} \cdot y$ ) plus $\ 20$ equals twice the original number ( $2 \cdot y$ ).
Let's write that as an equation: $\frac{1}{3} \cdot y + 20 = 2 \cdot y$ .

STEP 9

Let's get rid of the fraction!
Multiply everything by $\ 3$ : $3 \cdot (\frac{1}{3} \cdot y + 20) = 3 \cdot (2 \cdot y)$ .
This simplifies to $y + 60 = 6y$ .

STEP 10

Now, subtract $y$ from both sides: $60 = 6y - y$ , which means $60 = 5y$ .

STEP 11

Finally, divide both sides by $\ 5$ : $y = \frac{60}{5}$ , so $y = 12$ .
Our mystery number is $\ 12$ !

STEP 12

The two numbers that add up to 2024 and have a remainder of 16 when divided are 1004 and 1020.
The mystery number that becomes twice itself when you take one-third of it and add 20 is 12.

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Studdy Solution

STEP 1

STEP 2

1. Solve the division puzzle2. Solve the one-third plus 20 puzzle

STEP 3

Alright, let's tackle the first mystery!We know two numbers add up to **2024**, and when we divide the bigger one by the smaller one, we get a remainder of **16**.

STEP 4

Think of it like this: If it weren't for that pesky remainder, the bigger number would be a perfect multiple of the smaller number.So, let's subtract that remainder! 2024−16=2008\ 2024 - 16 = 2008 2024−16=2008.Now, we know that 2008\ 2008 2008 is perfectly divisible by the smaller number.

STEP 5

STEP 6

Subtract 16 16 16 from both sides: 2x=2024−16 2x = 2024 - 16 2x=2024−16, which means 2x=2008 2x = 2008 2x=2008.Now, divide both sides by 2\ 2 2: x=20082 x = \frac{2008}{2} x=22008​, so x=1004 x = 1004 x=1004.That's our smaller number!

STEP 7

The larger number is 2024−1004=1020\ 2024 - 1004 = 1020 2024−1004=1020.Let's check: 10201004=1\ \frac{1020}{1004} = 1 10041020​=1 with a remainder of 16\ 16 16.Perfect!

STEP 8

Let's call our mystery number y y y.The problem says one-third of the number (13⋅y \frac{1}{3} \cdot y 31​⋅y) plus 20\ 20 20 equals twice the original number (2⋅y 2 \cdot y 2⋅y).Let's write that as an equation: 13⋅y+20=2⋅y \frac{1}{3} \cdot y + 20 = 2 \cdot y 31​⋅y+20=2⋅y.

STEP 9

Let's get rid of the fraction!Multiply everything by 3\ 3 3: 3⋅(13⋅y+20)=3⋅(2⋅y) 3 \cdot (\frac{1}{3} \cdot y + 20) = 3 \cdot (2 \cdot y) 3⋅(31​⋅y+20)=3⋅(2⋅y).This simplifies to y+60=6y y + 60 = 6y y+60=6y.

STEP 10

Now, subtract y y y from both sides: 60=6y−y 60 = 6y - y 60=6y−y, which means 60=5y 60 = 5y 60=5y.

STEP 11

Finally, divide both sides by 5\ 5 5: y=605 y = \frac{60}{5} y=560​, so y=12 y = 12 y=12.Our mystery number is 12\ 12 12!

STEP 12

The two numbers that add up to 2024 and have a remainder of 16 when divided are **1004** and **1020**.The mystery number that becomes twice itself when you take one-third of it and add 20 is **12**.

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1. Solve the division puzzle
2. Solve the one-third plus 20 puzzle

Alright, let's tackle the first mystery!
We know two numbers add up to 2024, and when we divide the bigger one by the smaller one, we get a remainder of 16.

Think of it like this: If it weren't for that pesky remainder, the bigger number would be a perfect multiple of the smaller number.
So, let's subtract that remainder! $\ 2024 - 16 = 2008$ .
Now, we know that $\ 2008$ is perfectly divisible by the smaller number.

Subtract $16$ from both sides: $2x = 2024 - 16$ , which means $2x = 2008$ .
Now, divide both sides by $\ 2$ : $x = \frac{2008}{2}$ , so $x = 1004$ .
That's our smaller number!

The larger number is $\ 2024 - 1004 = 1020$ .
Let's check: $\ \frac{1020}{1004} = 1$ with a remainder of $\ 16$ .
Perfect!

Let's call our mystery number $y$ .
The problem says one-third of the number ( $\frac{1}{3} \cdot y$ ) plus $\ 20$ equals twice the original number ( $2 \cdot y$ ).
Let's write that as an equation: $\frac{1}{3} \cdot y + 20 = 2 \cdot y$ .

Let's get rid of the fraction!
Multiply everything by $\ 3$ : $3 \cdot (\frac{1}{3} \cdot y + 20) = 3 \cdot (2 \cdot y)$ .
This simplifies to $y + 60 = 6y$ .

Now, subtract $y$ from both sides: $60 = 6y - y$ , which means $60 = 5y$ .

Finally, divide both sides by $\ 5$ : $y = \frac{60}{5}$ , so $y = 12$ .
Our mystery number is $\ 12$ !

The two numbers that add up to 2024 and have a remainder of 16 when divided are 1004 and 1020.
The mystery number that becomes twice itself when you take one-third of it and add 20 is 12.