Associated Topics || Dr. Math Home || Search Dr. Math

### Number Puzzle: 26, 70

```
Date: 09/12/2001 at 01:45:44
From: Bryce  Aspell
Subject: Word problem

The sum of two numbers is 26; 4 times the first number plus 2 times
the second number equals 70. What are the two numbers?
```

```
Date: 09/12/2001 at 14:54:52
From: Doctor Ian
Subject: Re: Word problem

Hi Bryce,

Just for fun, let's try guessing that the two numbers are 12 and 14.

12 + 14 = 26              (good)

4*12 + 2*14 = 48 + 28

= 76          (too big)

We overshot on the second criterion. How can we bring that down a
little?

Well, notice that the first number gets multiplied by 4, while the
second number gets multiplied by only 2. So if we can make the first
number a little smaller, maybe we can get the second number down to
70:

10 + 16 = 26             (good)

4*10 + 2*16 = 40 + 32

= 72         (still too big)

We still overshot by a little. But we're clearly going in the right
direction. What about 8 and 18?

8 + 18 = 26              (good)

4*8 + 2*18 = 32 + 36

= 68          (too small)

So the first number has to be a little bigger. That's good! It means
that the answer has to be somewhere in between the numbers we've
tried.

You can keep going this way, or you can try to solve the problem
directly, by translating it into equations.  Let's try that:

1.  The sum of two numbers is 26:

If one of the numbers is N, then the other one has to be 26-N.
Do you see why?

2. Four times the first number (4 * N) plus two times the
second number (2*(26-N) equals 70:

4N + 2(26-N) = 70

Now you can try various values for N:

N=5  ->   4(5) + 2(26 - 5)   = 20 + 42 = 62       (too low)

N=6  ->   4(6) + 2(26 - 6)   = 24 + 40 = 64       (too low)

[the answer has to be in here somewhere]

N=10 ->   4(10) + 2(26 - 10) = 40 + 32 = 72       (too big)

Or you can use algebra to solve the equation directly:

4N + 2(26 - N) = 70

4N + 2(26) - 2(N) = 70

4N - 2N + 52 = 70

2N = 70 - 52

and so on.

There are often lots of different ways that you can solve any given
problem, so it's good to try to be aware of your options, and to keep
in mind that _any_ method that helps you solve the problem quickly is
the 'right' method... even if it's guessing, or making a graph, or
counting on your fingers, or moving pennies around on a desk.

As W.W. Sawyer wrote in his excellent book, _Vision in Elementary
Mathematics_,

Pupils ask 'Am I allowed to do this?' as if we were playing a game
with certain rules. A pupil is allowed to write anything that
is true, and not allowed to write anything untrue! These are
the only rules of mathematics.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Puzzles
Middle School Algebra
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search