Solving Proportions: Practice Problems and Equations

Solving Proportions

Proportions involve comparing two ratios with each other. For example, if we say "two is to four as three is to six", we can express this as a proportion: (2 / 4) = (3 / 6). This means that every time something described by the first part of the proportion happens once, it will happen twice when described by the second part.

To solve proportions, you follow these steps:

1. Identify the parts of the proportion. In our example, they are (2 / 4) and (3 / 6).

   • The numbers before the slashes are called the numerators. They tell us how many times something happens in one situation.
   • The numbers after the slash are called the denominators. They tell us how many times something happens in another situation.
   • To check if two quantities are equal, make sure both numerator and denominator are equal or related[(#_ftn1)].

2. Choose a variable. In our example, there isn't any explicit variable, so let's label the parts of the proportion as A, B, C, and D left to right from the top. Note that subscripts are used to distinguish between numerators or denominators in different proportions.

3. Set up equations based on the equation of proportionality. If you have (a / b) = (c / d), then multiply both sides by d, getting ad = bc. Alternatively, you can divide both sides by c, obtaining a/c = b/d. These rules apply even if the subscripts don't match.

Let's practice some problems using these steps:

A => B => C => D ====== ====== ====== 2 x 5 x 8 x 13 9 x 7 x 12 x 11

In this case, we have two sets of variables: {A, B} = {2, 5} and {C, D} = {13, 11}. We need to find values for x, y, z, and w such that PQRS forms a valid proportion, where P = A/B, Q = B/C, R = C/D, S = D/E, and T = E/F. The problem states that 13 + 11 = 2 * 5, which gives us T = F. Since all other values are given, the equation becomes T = S = R = Q = P = 1. Therefore, F = 2, E = 5, D = 13, C = 2, B = 5, and A = 2.

Given the information:

7 x T x 3 x K x L x 5 x M x N x J x I x H x G x F x E x D x C x B x A x Z x Y ====== ====== ====== ====== ====== ====== ====== ====== ====== ====== ====== Q x N x P x M x L x K x J x I x E x D x C x B x A x Z x Y

We have three sets of variables: {A, B} = {K, N}, {C, D} = {F, E}, and {G, H} = {Z, Y}. Let's solve for x and z instead of B and D. We get:

P + Q = M N + K = J 5 * F + T = L 5 * E + z = J 13 * F + 11 * E + T + z = 3 * K + 2 * N 5 * G + w = 3 * Z 5 * H + v = 3 * Y

Since there are more variables than equations, we cannot uniquely determine all the variables. However, we can find the values of x and z once we have enough information to complete the equation:

13 * F + 11 * E + T + z = 3 * K + 2 * N

We know that F = 2, so we can substitute this into the equation and solve for z:

13 * 2 + 11 * E + T + z = 3 * K + 2 * N
2 * 13 + 11 * E + T + z = 3 * K + 2 * N
26 + 11 * E + T + z = 3 * K + 2 * N

Now we need to determine the values of E, T, and z. Without additional information, this cannot be done.