Ratio and Algebra

Write the ratio equality as an equality of fractions.

1. The ratio gives \frac{x}{12}=\frac23.
2. Multiply by 12: x=12\times\frac23=8.

Use the known first quantity to find the value of one ratio part.

1. The first 5 parts represent 15, so each part represents 3.
2. The second quantity is 4 parts, giving y=12.

The first quantity is the first ratio part divided by the second, multiplied by the second quantity.

1. The ratio means \frac ab=\frac38.
2. Multiply both sides by b, giving a=\frac{3b}{8}.

Find the value of the entire first ratio term before solving for the unknown.

1. The first term is x+1=9\times\frac23=6.
2. Subtract 1 to obtain x=5.

First find the value of the term containing three times the unknown.

1. The ratio gives 3x=20\times\frac9{10}=18.
2. Divide by 3, giving x=6.

The total corresponds to the sum of the ratio parts.

1. The total T is 4+7=11 equal parts.
2. Since a is four parts, a=\frac4{11}T=\frac{4T}{11}.

Cross-multiply to turn the ratio into a linear equation.

1. The ratio gives 3(x-2)=x+4.
2. Expand and collect: 3x-6=x+4, so 2x=10.
3. Thus x=5, which satisfies x>2.

Cross-multiply and keep the bracket around the full first ratio term.

1. The equality of ratios gives 2(2x+3)=5x.
2. Expanding gives 4x+6=5x.
3. Subtract 4x to obtain x=6.

Represent both quantities using the same scale factor.

1. Write p=2k and q=5k, with k>0.
2. The numerator becomes 7k and the denominator becomes 3k.
3. Cancelling the common factor gives \frac73.

Collect the p terms on one side and the q terms on the other.

1. Collect like terms: 3q+2q=7p-4p, so 5q=3p.
2. Divide both sides by 3q: \frac pq=\frac53.
3. So p:q=5:3. Check: p=5 and q=3 give 29 on both sides.
4. The tempting 3:5 comes from copying the numbers in 3p=5q. In 3p=5q, p is the larger number, so p:q=5:3.

Cross-multiply to form an equation involving both bracketed expressions.

1. The ratio gives 4(x+5)=3(2x-1).
2. Expand: 4x+20=6x-3.
3. Rearranging gives 23=2x, so x=\frac{23}{2}. Both ratio terms are positive.

Write the ratio as a fraction equality, then cross-multiply.

1. The ratio gives 3(3x+2)=2(5x-4).
2. Expanding gives 9x+6=10x-8.
3. Hence x=14. The quantities are then 44 and 66, in the required ratio.

Combine the two ratios using a common number of parts for the middle quantity.

1. Scale the ratios to a:b=8:12 and b:c=12:21.
2. Therefore a:c=8:21.
3. The expression for c is \frac{21a}{8}.

Use a shared multiplier and translate the difference into an equation.

1. Write x=5k and y=2k.
2. The difference gives 3k=18, so k=6.
3. Hence x=5\times6=30.

Cross-multiply, then collect the b terms on one side.

1. Write the ratio as equal fractions: \frac{2a+b}{a-b}=\frac72.
2. Cross-multiply: 2(2a+b)=7(a-b).
3. Expand the brackets: 4a+2b=7a-7b.
4. Collect the b terms on one side and the a terms on the other: 9b=3a.
5. Divide by 9: b=\frac{3a}{9}=\frac a3.

Cross-multiply to form a quadratic, then reject any solution that violates the domain.

1. Cross-multiplication gives 4(x+3)=x^2.
2. Rearrange to get x^2-4x-12=0.
3. Factorise: (x-6)(x+2)=0, so x=6 or x=-2.
4. The condition x>0 leaves x=6.

Use a shared multiplier for the original numbers, then form the ratio after the addition.

1. Write a=2k and b=3k, with k>0.
2. The changed ratio gives \frac{2k+4}{3k+4}=\frac34.
3. Cross-multiply: 8k+16=9k+12, so k=4.
4. Therefore a=2\times4=8.

Express all three quantities with a shared multiplier before substituting into the fraction.

1. Double the second ratio to get y:z=4:10, so x:y:z=3:4:10.
2. Write x=3k, y=4k and z=10k.
3. The required fraction is \frac{6k+10k}{3k+4k}=\frac{16k}{7k}.
4. Cancelling k gives \frac{16}{7}.

The product of the two ratio expressions will contain the square of their common multiplier.

1. Write the numbers as 2k and 5k, with k>0.
2. The product gives 10k^2=90, so k^2=9.
3. The positive multiplier is k=3.
4. The larger number is 5\times3=15.

Move every term to one side and factorise.

1. Make one side zero: 3x^2+5xy-2y^2=0.
2. This factorises in the same way as 3t^2+5t-2=(3t-1)(t+2), giving (3x-y)(x+2y)=0.
3. x and y are both positive, so x+2y cannot be 0. So 3x-y=0, which gives y=3x.
4. So x:y=x:3x=1:3.
5. The option 2:1 comes from x+2y=0, which would need x=-2y: impossible for positive numbers. 3:1 is the ratio the wrong way round.