Math  /  Algebra

Question27. Хоёр оронтой тоог нэг тоонд хуваахад үлдэгдэл гарав. Хэрэв үлдэгдэл ноогдвороос 3 дахин бага, ноогдвор нь хуваагчаас 4 дахин бага бол хуваагдагчийг ол.

Studdy Solution

STEP 1

1. We have a two-digit number that, when divided by another number, leaves a remainder.
2. The remainder is three times smaller than the quotient.
3. The quotient is four times smaller than the divisor.
4. We need to find the dividend (the two-digit number).

STEP 2

1. Define variables for the unknowns.
2. Set up equations based on the problem's conditions.
3. Solve the equations to find the dividend.

STEP 3

Define variables for the unknowns.
Let: - D D be the dividend (the two-digit number). - d d be the divisor. - q q be the quotient. - r r be the remainder.

STEP 4

Set up equations based on the problem's conditions.
From the problem, we have the following conditions:
1. r=q3 r = \frac{q}{3} (The remainder is three times smaller than the quotient).
2. q=d4 q = \frac{d}{4} (The quotient is four times smaller than the divisor).
3. The relationship between dividend, divisor, quotient, and remainder is given by: D=dq+r D = dq + r

STEP 5

Solve the equations to find the dividend.
Substitute r=q3 r = \frac{q}{3} and q=d4 q = \frac{d}{4} into the equation D=dq+r D = dq + r :
D=d(d4)+d43 D = d \left(\frac{d}{4}\right) + \frac{\frac{d}{4}}{3}
Simplify the equation:
D=d24+d12 D = \frac{d^2}{4} + \frac{d}{12}
Combine terms over a common denominator:
D=3d212+d12 D = \frac{3d^2}{12} + \frac{d}{12}
D=3d2+d12 D = \frac{3d^2 + d}{12}
Since D D is a two-digit number, we need to find integer values for d d that make D D a two-digit number.

STEP 6

Test integer values for d d to find a valid two-digit D D .
Let's try d=12 d = 12 :
D=3(12)2+1212 D = \frac{3(12)^2 + 12}{12}
D=3(144)+1212 D = \frac{3(144) + 12}{12}
D=432+1212 D = \frac{432 + 12}{12}
D=44412 D = \frac{444}{12}
D=37 D = 37
Since 37 is a two-digit number, this is a valid solution.
The dividend is:
37 \boxed{37}

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