Math / Algebra

Question\$e number which is added to each of the numbers 1,3 and 6 to become in continued proportion is

Studdy Solution

STEP 1

1. We are looking for a number $x$ that, when added to each of the numbers 1, 3, and 6, makes them form a continued proportion.
2. A continued proportion means that the ratio of the first two terms is equal to the ratio of the last two terms.

STEP 2

1. Define the variable and set up the proportion.
2. Write the equation for the continued proportion.
3. Solve the equation for the unknown number.

STEP 3

Define the variable. Let $x$ be the number to be added to each of the numbers 1, 3, and 6.

STEP 4

Write the equation for the continued proportion. The numbers $1 + x$ , $3 + x$ , and $6 + x$ should be in continued proportion, which means:
$\frac{1 + x}{3 + x} = \frac{3 + x}{6 + x}$

STEP 5

Cross-multiply to solve the equation:
$(1 + x)(6 + x) = (3 + x)(3 + x)$
Expand both sides:
$6 + x + 6x + x^2 = 9 + 6x + x^2$
Simplify the equation by subtracting $x^2$ and $6x$ from both sides:
$6 + x = 9$

STEP 6

Isolate $x$ by subtracting 6 from both sides:
$x = 9 - 6$ $x = 3$
The number to be added is:
$\boxed{3}$

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Question\$e number which is added to each of the numbers 1,3 and 6 to become in continued proportion is

Studdy Solution

STEP 1

1. We are looking for a number $x$ that, when added to each of the numbers 1, 3, and 6, makes them form a continued proportion.
2. A continued proportion means that the ratio of the first two terms is equal to the ratio of the last two terms.

STEP 2

1. Define the variable and set up the proportion.
2. Write the equation for the continued proportion.
3. Solve the equation for the unknown number.

STEP 3

Define the variable. Let $x$ be the number to be added to each of the numbers 1, 3, and 6.

STEP 4

Write the equation for the continued proportion. The numbers $1 + x$ , $3 + x$ , and $6 + x$ should be in continued proportion, which means:
$\frac{1 + x}{3 + x} = \frac{3 + x}{6 + x}$

STEP 5

Cross-multiply to solve the equation:
$(1 + x)(6 + x) = (3 + x)(3 + x)$
Expand both sides:
$6 + x + 6x + x^2 = 9 + 6x + x^2$
Simplify the equation by subtracting $x^2$ and $6x$ from both sides:
$6 + x = 9$

STEP 6

Isolate $x$ by subtracting 6 from both sides:
$x = 9 - 6$ $x = 3$
The number to be added is:
$\boxed{3}$

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Question\$e number which is added to each of the numbers 1,3 and 6 to become in continued proportion is

Studdy Solution

STEP 1

1. We are looking for a number x x x that, when added to each of the numbers 1, 3, and 6, makes them form a continued proportion.2. A continued proportion means that the ratio of the first two terms is equal to the ratio of the last two terms.

STEP 2

1. Define the variable and set up the proportion.2. Write the equation for the continued proportion.3. Solve the equation for the unknown number.

STEP 3

Define the variable. Let x x x be the number to be added to each of the numbers 1, 3, and 6.

STEP 4

Write the equation for the continued proportion. The numbers 1+x 1 + x 1+x, 3+x 3 + x 3+x, and 6+x 6 + x 6+x should be in continued proportion, which means:1+x3+x=3+x6+x\frac{1 + x}{3 + x} = \frac{3 + x}{6 + x}3+x1+x​=6+x3+x​

STEP 5

Cross-multiply to solve the equation:(1+x)(6+x)=(3+x)(3+x)(1 + x)(6 + x) = (3 + x)(3 + x)(1+x)(6+x)=(3+x)(3+x)Expand both sides:6+x+6x+x2=9+6x+x26 + x + 6x + x^2 = 9 + 6x + x^26+x+6x+x2=9+6x+x2Simplify the equation by subtracting x2 x^2 x2 and 6x 6x 6x from both sides:6+x=96 + x = 96+x=9

STEP 6

Isolate x x x by subtracting 6 from both sides:x=9−6x = 9 - 6x=9−6 x=3x = 3x=3The number to be added is:3 \boxed{3} 3​

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1. We are looking for a number $x$ that, when added to each of the numbers 1, 3, and 6, makes them form a continued proportion.
2. A continued proportion means that the ratio of the first two terms is equal to the ratio of the last two terms.

1. Define the variable and set up the proportion.
2. Write the equation for the continued proportion.
3. Solve the equation for the unknown number.

Define the variable. Let $x$ be the number to be added to each of the numbers 1, 3, and 6.

Write the equation for the continued proportion. The numbers $1 + x$ , $3 + x$ , and $6 + x$ should be in continued proportion, which means:
$\frac{1 + x}{3 + x} = \frac{3 + x}{6 + x}$

Cross-multiply to solve the equation:
$(1 + x)(6 + x) = (3 + x)(3 + x)$
Expand both sides:
$6 + x + 6x + x^2 = 9 + 6x + x^2$
Simplify the equation by subtracting $x^2$ and $6x$ from both sides:
$6 + x = 9$

Isolate $x$ by subtracting 6 from both sides:
$x = 9 - 6$ $x = 3$
The number to be added is:
$\boxed{3}$